Glossary: A

Definitions in this project whose names start with A. The whole glossary has the other sections.

A · B · C · D · E · F · G · H · I · J · K · L · M · N · O · P · Q · R · S · T · U · V · W · X · Y · Z · Other

A def: $A_n = \operatorname{Re} Z_n$. · Arxiv.«2607.05739».A
A def: def A (n : ℕ) : Set (ℤ × ℤ × ℤ) · CongruentNumber.A
A def: Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$. · Erdos1073.A
A def: The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence. · Erdos1101.A
A def: The smallest set of natural numbers which contains 1 and is closed under the operations x ↦ 2x + 1, x ↦ 3x + 1 and x ↦ 6x + 1. · Erdos1134.A
A abbrev: Let $A(x)$ denote the set of indices $n$ for which $s_n \leq x$. · Erdos145.A
A def: The set $A$ of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$. · Erdos292.A
A def: $A(N)$ is the maximal cardinality of $A\subseteq \{1,\ldots,N\}$ such that $\sum_{n\in S}\frac{1}{n}\neq 1$ for all $S\subseteq A$. · Erdos300.A
A def: The set $\{k(m-k) : 1\leq k\leq m/2\}$, where m / 2 is ℕ floor division. · Erdos443.A
A abbrev: Let $A(x)$ count the number of amicable pairs $1\leq a\leq b\leq x$. · Erdos830.A
A def: For an infinite sequence $x_1, x_2, \ldots \in (0, 1)$, define $$A_k = \limsup_{n \to \infty} \left\lvert \sum_{j \le n} e(k x_j) \right\rvert,$$ where $e(x) = e^{2\pi i x}$. · Erdos987.A
A def: def A : ℝ · MovingSofa.GerversSofa.A
A def: The predicate defining whether $k$ belongs to the sequence. · OeisA108569.A
A def: The condition for a number $n$ to be in the sequence. It satisfies $0 < n$ and its perfect deficiency is $\le 10$. · OeisA108864.A
A def: The predicate that n can be written as $x + y$ with $x,y >0$ such that $2^x + y$ is prime · OeisA231201.A
A def: The predicate that n can be written as $x + y$ with $x, y > 0$ such that both $x + ny$ and $x^2 + ny^2$ are prime. · OeisA232174.A
A def: The predicate that $p$ has a primitive root $0 < g < p$ of the form $k^2 + 1$, where $k$ is an integer. · OeisA239957.A
A def: A natural number is in A002407 when it is prime and is the difference of two consecutive positive cubes. The addition equality avoids truncated subtraction in ℕ. · OeisA2407.A
A def: The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers such that $x^4 + 1680 y^3 z$ is a square. · OeisA280831.A
A def: The predicate that n can be written as $x^2 + y^2 + z^2 + w^2$ with $x, y, z, w$ nonnegative integers, $z \leq w$, such that both $x$ and $x + 24y$ are squares. · OeisA281976.A
A def: The predicate that n can be written as $x(x+1)/2 + y(3y+1)/2 + z(5z+1)/2$ for nonnegative integers $x, y, z$. · OeisA287616.A
A def: The predicate that $n$ can be written as $a^2 + b^2 + \binom{2c+1}{c} + \binom{2d+1}{d}$ for nonnegative integers $a, b, c, d$. · OeisA303639.A
A def: The predicate that n can be written as $a^2 + b^2 + 3^c + 5^d$ for nonnegative integers. · OeisA303656.A
A def: The predicate that n can be written as $\binom{w+2}{2} + \binom{x+3}{4} + \binom{y+5}{6} + \binom{z+7}{8}$ for nonnegative integers $w, x, y, z$. · OeisA306477.A
A def: The predicate that n can be written as $(2^a \cdot 3^b)^2 + (2^c \cdot 5^d)^2 + x^2 + y^2$ for nonnegative integers $a, b, c, d, x, y$. · OeisA308734.A
A def: A prime $p$ is in A003625 if $p \equiv 3, 5, \text{ or } 6 \pmod 7$. · OeisA3625.A
A def: A positive integer $n$ is practical if every $m \le n$ can be represented as a sum of distinct divisors of $n$. · OeisA5153.A
A def: A composite number $n$ is in the sequence A56777 if it satisfies both $\varphi(n+12) = \varphi(n) + 12$ and $\sigma(n+12) = \sigma(n) + 12$. · OeisA56777.A
A def: A number $n$ is in the sequence A063880 if $\sigma(n) = 2 \cdot \text{usigma}(n)$. · OeisA63880.A
A def: A number $k$ is in the sequence A067720 if $\varphi(k^2 + 1) = k \cdot \varphi(k + 1)$. · OeisA67720.A
A def: The gcd of the binomial coefficients $\binom{2k}{k}, \binom{3k}{k}, \dots, \binom{(k+1)k}{k} = 1$. · OeisA80170.A
A def: Primes in A81091 have exactly $3$ set bits in binary representation ($2^n + 2^i + 1$). · OeisA81091.A
A abbrev: The leading constant $\zeta(3/2)/\zeta(3) = 2.173\ldots$ · PowerfulNumbersDensity.A
a def: The infinite, strictly increasing sequence $\{a_0, a_1, \dots\}$ of integers whose prime factors all belong to $P$. · Erdos269.a
a def: $a_n$ is defined by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}$. · Erdos291.a
a def: The sequence of the problem: $a_1 = 1$ and $a_{n+1} = \lfloor \sqrt{2}(a_n + 1/2) \rfloor$ for $n \geq 1$. The value $a_0$ is not used. · Erdos482.a
a def: The primary defining sequence a, which is the expansion of the generating function $(1+x)(3+x)/(1+6x^2+x^4)$. It satisfies the recurrence $a(n) = -6 a(n-2) - a(n-4)$ for $n \ge 4$. · OeisA100434.a
a def: The primary defining sequence a. · OeisA100474.a
a def: The primary defining sequence a. a n is the Prime-th recurrence with reversal at each step. $a(n) = \operatorname{reversal}(p_{a(n-1)})$ with $a(0)=1$, where $p_k$ is the $k$-th prime number (i.e., $p_1=2, p_2=3, \dots$). · OeisA100475.a
a def: The primary defining sequence a. Pentanacci $\pi$ sequence: $a(1)=a(2)=a(3)=a(4)=a(5)=1$; for $n>5$, $a(n) = \pi(\sum_{j=1}^5 a(n-j))$ where $\pi = A000720$. Note on indices: for $n \ge 0$, $a(n)$ corresponds to $A_{n+1}$ in the OEIS … · OeisA100478.a
a def: a n is the first iteration of $f(n) = n + \text{sum of the digits of } n$ that is a multiple of $n$. $a(n) = 0$ if no such number exists. · OeisA100800.a
a def: The primary defining sequence a. $a(n)$ is the least $k$ such that all of $k, 2k+1, 3k+2, \ldots, nk+n-1$ are primes, or $0$ if no such $k$ is found. · OeisA101779.a
a def: The primary defining sequence a. $a(n)$ is defined by the recurrence relation $a(1)=1$, and for $n>1$, $a(n) = a(n-1) \operatorname{XOR} (a(n-1) + n)$, where $\operatorname{XOR}$ is the bitwise exclusive-or operator (^^^). · OeisA102371.a
a def: The primary defining sequence a. a n is the floor of the sum of all division remainders $\{n/k\}$, with $k=1,\dots,n$. $$a(n) = \left\lfloor \sum_{k=1}^n \left\{ \frac{n}{k} \right\} \right\rfloor$$ · OeisA102722.a
a def: The primary defining sequence a. $a(0)=1$, $a(n) = a(n-1)^2 + 2$. · OeisA102847.a
a def: The primary defining sequence a. a n is the number of decompositions of 2n+1 into 2p+q, where p and q are both odd primes. · OeisA103151.a
a def: The sequence $a(n)$ satisfies the linear recurrence relation: $$a(n) = 3a(n-1) - 4a(n-2) + 2a(n-3) - a(n-4)$$ with initial terms $a(0)=0, a(1)=1, a(2)=1, a(3)=0$. The sequence takes values in $\mathbb{Z}$. · OeisA103311.a
a def: The primary defining sequence a. $a(n)$ is defined by the recurrence relation $a(n) = 3 a(n-1) + a(n-2) - 3 a(n-3)$ with initial terms $a(0)=1, a(1)=3, a(2)=5$. · OeisA103425.a
a def: The primary defining sequence a. a n is the smallest power with base $>1$ and exponent $n$ whose decimal representation doesn't contain the digit 0. $$a(n) = (\min { b \in \mathbb{N} \mid b > 1, \text{decimal representation of } b^n … · OeisA103662.a
a def: The primary defining sequence a. $a(n) = [x^{2n}] \left(\frac{1 + x}{1 - x}\right)^n$, given by the combinatorial identity: $a(n) = \sum_{k = 0}^n \binom{n}{k} \binom{2n+k-1}{n-1}$ with $a(0) = 1$. · OeisA103885.a
a def: The primary defining sequence a. a n is the number of zeros in the ternary representation of $2^n$. · OeisA104320.a
a def: The primary defining sequence a. Array read by upward antidiagonals: row $n$ ($n \ge 0$) contains the numbers $m^2 - n^2$, $m \ge n+1$. The array entry $T(i, j)$ is $(i+j+1)^2 - i^2$. The sequence $a(n)$ is generated by reading $T(i, j)$ … · OeisA105020.a
a def: The primary defining sequence a. a n is defined by the formula: $$a(n) = n - \sum_{k \ge 0, 2^{k+1} \le n, n \equiv k \pmod{2^{k+1}}} 2^{k+1}$$ · OeisA105033.a
a def: The primary defining sequence a. $a(1) = 393$; for $n > 1$, $a(n)$ = $a(n-1)$ + 1 + sum of distinct prime factors of $a(n-1)$ that are < $a(n-1)$. · OeisA105210.a
a def: The primary defining sequence a. $a(n) = 1$ if exactly 5 Fibonacci numbers exist with exactly $n$ digits, otherwise $0$. That is, $a(n) = 1$ if the number of indices $k \in \mathbb{N}$ such that $10^{n-1} \le \mathrm{fib}(k) < 10^n$ is 5. · OeisA105565.a
a def: The primary defining sequence a. $a(n)$ is the sum of primes from the $n$-th prime to the $2n$-th prime, with $p_1=2$. For $n=0$, we define $a(0) = 0$. · OeisA105720.a
a def: The primary defining sequence a. a n is the imaginary part of $\prod_{k=0}^n (1 + k \cdot i)$, where $i = \sqrt{-1}$. · OeisA105751.a
a def: The primary defining sequence a. $a(n)$ is the Fibonacci-Collatz sequence: $a(1)=1, a(2)=2$; for $n > 2$, let $\mathrm{fib} = a(n-1) + a(n-2)$; if $\mathrm{fib}$ is odd then $a(n) = 3 \cdot \mathrm{fib} + 1$ else $a(n) = \mathrm{fib}/2$. · OeisA105801.a
a def: The primary defining sequence a. a n is the sum of squares of nonacci numbers (Fibonacci 9-step numbers). The official sequence matches the sum up to n + 1 of the standard nonacci numbers to align with the offset. · OeisA107247.a
a def: Catalan numbers $C(n) = \frac{1}{n+1}\binom{2n}{n}$. · OeisA108.a
a def: The primary defining sequence a. $a(n) = \sum_{k=0}^n \binom{n+k-1}{k} F(n-k+1)$, where $F(m)$ is the $m$-th Fibonacci number. · OeisA108081.a
a def: The primary defining sequence a. Riesel problem: let $k=2n-1$; then $a(n)$ is the smallest $m \ge 1$ such that $k \cdot 2^m-1$ is prime, or $-1$ if no such prime exists. · OeisA108129.a
a def: The primary defining sequence a. a n is defined as $16n^2 + 1$. · OeisA108211.a
a def: The primary defining sequence a. a n is the digital sum of the Fermat number $2^{2^n} + 1$. · OeisA108301.a
a def: The primary defining sequence a. a n is the $n$-th term of the expansion of $(3x+1)/(1-3x-3x^2)$, satisfying $a(0)=1, a(1)=6, a(n)=3a(n-1)+3a(n-2)$. · OeisA108306.a
a def: The primary defining sequence a. a n is the $(n+1)$-th positive integer $k$ such that $\phi(k) = \phi(k + \phi(k))$. · OeisA108569.a
a def: The primary defining sequence a. $a(n)$ is the $n$-th number (0-indexed) such that its perfect deficiency is $\le 10$. · OeisA108864.a
a def: The primary defining sequence a. a n is the numerator of $\sum_{k=1}^n \frac{2^k}{k}$. · OeisA108866.a
a def: The primary defining sequence a. $a(n)$ is the numerator of $\binom{6n-2}{2n} / \left(2 \binom{4n-1}{2n}\right)$. · OeisA109074.a
a def: The primary defining sequence a. $a(n)$ is the natural number whose decimal digits are given by the binary string of length $p_n$ where the $k$-th digit is $1$ if $k$ is prime and $0$ otherwise, with leading zeros dropped. · OeisA109227.a
a def: The primary defining sequence a. $a(1)=1$; thereafter, $a(2n)=a(n)$, $a(2n+1)$ is the smallest positive number such that $|a(2n+1)-a(2n-1)|=a(n)$. · OeisA109671.a
a def: $a(1) = 2$; $a(2n)$ = lcm of all previous terms + 1; $a(2n+1)$ = lcm of all previous terms - 1. The sequence is correctly generated by the recurrence relation $a(1)=2$, $a(2)=3$, and for $n \ge 3$: $$a(n) = a(n-1)^2 + (-1)^n a(n-1) + … · OeisA109845.a
a def: $a(n)$ is the greatest prime of the form $k(n-k)+1$, where $k$ can take values from $1$ to $\lfloor n/2 \rfloor$. $a(n) = 0$ if no such prime exists. · OeisA109905.a
a def: $a(n)$ = greatest prime of the form $k(n-k)-1$, or $0$ if no such prime exists. · OeisA109908.a
a def: $a(n)$ is the number of primes of the form $k(n-k)-1$. This counts the number of distinct prime values in the set $\{k(n-k)-1 | 1 \le k < n\}$. · OeisA109909.a
a def: The primary defining sequence a. $a(n)$ is the number of symbols '*' and '^' to write the canonical prime factorization of $n$. · OeisA110475.a
a def: The primary defining sequence a. $a(n) = \frac{\operatorname{lcm}_{k=1}^n k}{\operatorname{den}(H_n)}$ · OeisA110566.a
a def: The primary defining sequence a. $a(n)$ is the smallest $m > 0$ such that there are no primes between $n \cdot m$ and $n \cdot (m+1)$ inclusive. · OeisA110835.a
a def: The primary defining sequence a. $a(n)$ is $\mathrm{prime}(2n+2) - \mathrm{prime}(2n+1) - \mathrm{prime}(2n) + \mathrm{prime}(2n-1)$. · OeisA110854.a
a def: a n is the integer part of $\mathrm{prime}(n)/\pi(n)$. Here $\mathrm{prime}(n)$ is the $n$-th prime number, and $\pi(n)$ is the prime-counting function. The sequence is defined for $n \ge 2$. · OeisA111114.a
a def: a n is the number of refactorable numbers $\le 10^n$. A number $k$ is refactorable if its number of divisors, $\tau(k)$, divides $k$. · OeisA111291.a
a def: a n is the $n$-th term of the sequence related to NOR bracketings. · OeisA112521.a
a def: a n is the generalized Stern sequence, defined by the recurrence relations: $a(2n+1) = a(n)$ and $a(2n) = a(n)$ + $a(n-2)$ with $a(0) = 1$, $a(1) = 1$ and $a(n) = 0$ for $n \le -1$. · OeisA112970.a
a def: a n is the {Number of digits of n} raised to the power of {the sum of the digits of n}. · OeisA113010.a
a def: a n is the (Number of digits of n) raised to the power of (the digital root of n), with appropriate adjustments for $n=0$. · OeisA113019.a
a def: a n is the smallest number m such that $2^n - m$ and $2^n + m$ are primes. Computed by checking elements from $0$ to $2^n$, as any larger $m$ would make $2^n - m = 0$ which is not prime. · OeisA113213.a
a def: a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 64 a(n-3) + 256 a(n-4)$ with initial values $-1, 4, 32, 64$. · OeisA113250.a
a def: a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 144 a(n-3) + 1296 a(n-4)$ with initial values $-1, 4, 92, 784$. · OeisA113252.a
a def: The sequence $a(n)$ is defined by the initial conditions $a(0)=-1, a(1)=4, a(2)=176, a(3)=3136$, and the linear recurrence relation $a(n) = -4 * a (n-1) + 256 * a (n-3) + 4096 * a (n-4)$ for $n \ge 4$. · OeisA113254.a
a def: a n is the sequence defined by the linear recurrence $a(n) = -4 a(n-1) + 324 a(n-3) + 6561 a(n-4)$ with initial values $-1, 4, 227, 5329$. · OeisA113255.a
a def: a n is the ascending descending base exponent transform of squares · OeisA113257.a
a def: The primary defining sequence a. $a(n)$ is the ascending descending base exponent transform of factorials. $$a(n) = \sum_{i = 1}^n (i!) ^ {(n-i+1)!}$$ · OeisA113258.a
a def: The primary defining sequence a. $a(n)$ is the ascending descending base exponent transform of $2^n$. $$a(n) = \sum_{i=0}^n 2^{i \cdot 2^{n-i}}$$ · OeisA113271.a
a def: The primary defining sequence a. $a(n)$ is the number of prime powers $q<=n$ such that also $q+2$ is a prime power. $$a(n) = \operatorname{card} \{q \in \mathbb{N} \mid 1 \le q \le n \land P(q) \land P(q+2) \}$$ · OeisA113609.a
a def: The primary defining sequence a. $a(n)$ is the difference between first odd semiprime > $2^n$ and $2^n$. $$a(n) = \min \{s \mid s > 2^n \text{ and } s \text{ is an odd semiprime}\} - 2^n$$ · OeisA114137.a
a def: The primary defining sequence a. $a(n)$ is the largest odd divisor of $a(n-1) + \textrm{prime}(n)$. · OeisA114216.a
a def: The primary defining sequence a. Numerator of $\zeta(4n)/\zeta(2n)^2$ (with $a(0)=2$ instead of $-2$). · OeisA114362.a
a def: The primary defining sequence a. $a(n) = 2^{2^n}$. · OeisA1146.a
a def: The primary defining sequence a. Each term is previous term plus floor of harmonic mean of two previous terms. · OeisA114831.a
a def: The primary defining sequence a. Partial sums of $\binom{2n}{n}^2$. · OeisA115257.a
a def: The primary defining sequence a. $a(n) = \#\{k \in \mathbb{N} \mid 1 \le k \le 10^n \land (k^2 + 3k + 1) \text{ is prime} \}.$ · OeisA115366.a
a def: a n is the integer whose decimal digits are the first $n+1$ decimal digits of $\pi$. · OeisA11545.a
a def: a n is the sum of squares of divisors of $n$. · OeisA1157.a
a def: a n is the sum of $3^j + (-2)^j$ for j from 1 to n. · OeisA116150.a
a def: a n is the determinant of a 2x2 matrix of non-overlapping blocks of 4 consecutive primes. · OeisA117027.a
a def: Number of primes in the $n$-th row of $T(n, k) = k^2 - k + p_n$ for $1 \le k \le n$. · OeisA117531.a
a def: Least $k > 0$ such that $|\Phi_k(n)|$ is prime, or $0$ if no such $k$ exists. · OeisA117545.a
a def: $a(n) = 2^{2^n} + 2^n - 1$. · OeisA119563.a
a def: Least $k \ge 1$ such that $2 \cdot n^k - 1$ is prime, or $0$ if no such $k$ exists. · OeisA119591.a
a def: Half-Fibonacci sequence starting with $1, 3$. · OeisA120424.a
a def: Prime gaps: differences between consecutive primes. · OeisA1223.a
a def: $a(n) = \frac{\prod_{j=1}^n \prod_{k=1}^n \gcd(j,k)}{\prod_{k=1}^n (\lfloor n/k \rfloor!)^k}$. · OeisA129365.a
a def: $a(n) = \sum_{i=0}^{n-1} \mathrm{signWeight}(p_i)$ where $p_i$ is the $i$-th prime. · OeisA130911.a
a def: $a(n) = x(n+1)/x(n) - 2$ for $n \ge 1$, and $a(0) = 0$. · OeisA135508.a
a def: The $n$-th lesser twin prime, with $a(0) = 0$. · OeisA1359.a
a def: Number of Abelian cubes of length $3n$ over an alphabet of size 3. · OeisA141057.a
a def: The sequence $a(n) = \mathrm{round}(\mathrm{round}(\sqrt{n!})/|(\mathrm{round}(\sqrt{n!}))^2 - n!|)$. · OeisA145355.a
a def: The sequence $a(n) = \mathrm{A006577}(n+1) - \mathrm{A006577}(n)$ for $n > 0$, or none if either $n$ or $n+1$ does not terminate. · OeisA153330.a
a def: Number of representations of $2n - 1$ as $p + 2^x + 7 \cdot 2^y$ with $p \equiv 5 \pmod 6$. · OeisA157225.a
a def: Number of representations of $2n - 1$ as $p + 2^x + 11 \cdot 2^y$ with $p \equiv 1 \pmod 6$. · OeisA157237.a
a def: $a(n)$ is the smallest natural number $m \ge 1$ such that $n^3 + m^3 + 1$ is prime, or none if no such $m$ exists. · OeisA159829.a
a def: Number of representations of $n$ as sum of square, pentagonal, and hexagonal numbers. · OeisA160324.a
a def: Defining recurrence for $a(n)$. · OeisA166944.a
a def: Chua's sequence, extended by $a(0) = 1$. · OeisA167604.a
a def: Smallest index $k > n$ such that $S(n) \mid S(k)$. · OeisA167918.a
a def: $a(n)$ is the denominator of the sum $$\sum_{i=1}^n \frac{1}{i} \binom{2n-i-1}{i-1}$$ · OeisA175386.a
a def: Rational recurrence sequence $a(n)$. · OeisA176477.a
a def: Denominator of sum of reciprocals of divisors of $n$. · OeisA17666.a
a def: The sequence $a(n) = \sum_{k=0}^n (-4)^k \binom{n}{k}^2 \binom{n-k}{k}^2$. · OeisA179524.a
a def: The sequence $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n-k}{k}^2 (-16)^k$. · OeisA179537.a
a def: Difference of sums of digits of $n$ in ternary and in binary. · OeisA180017.a
a def: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n-k}{k}^4$. · OeisA181546.a
a def: The sequence of squares of double factorials: $a(n) = ((2n-1)!!)^2$. · OeisA1818.a
a def: $a(n) = \mathrm{prime}(n) \cdot \mathrm{prime}(n+1) \bmod \mathrm{prime}(n+2)$. · OeisA182126.a
a def: Defining recurrence for $a(n)$. · OeisA182510.a
a def: Number of odd primes $p \in (n^2, (n+1)^2)$ with $(n/p) = 1$. · OeisA185150.a
a def: The sequence $a(n) = n! [x^n] \prod_{k=1}^n (1 - x^k / k!)$. · OeisA185895.a
a def: Size of the smallest subset $S$ of $T = \{1,2,3,\dots,n\}$ such that $S \cdot S$ contains $T$, where $S \cdot S$ is the set of all products of elements of $S$. · OeisA194806.a
a def: Integral factorial ratio sequence: $$a(n) = \frac{(30n)! n!}{(15n)! (10n)! (6n)!}$$ · OeisA211417.a
a def: For even $n$, $a(n+2)$ is the greatest integer such that $a(n+2)/a(n+1) < a(n+1)/a(n)$; for odd $n$, the least integer such that $a(n+2)/a(n+1) > a(n+1)/a(n)$; $a(0) = 4, a(1) = 16$. · OeisA22030.a
a def: The number of squares modulo $n$. · OeisA224.a
a def: a: $a(n) = \text{least } k \text{ such that } \sqrt{k^2 \operatorname{XOR} (k+1)^2} = 2n+1$, $a(n) = -1 \text{ if there is no such } k$. This is equivalent to finding the smallest $k \in \mathbb{N}$ such that $k^2 \oplus (k+1)^2 = … · OeisA224515.a
a def: Expansion of $(2+3x+2x^2+2x^3+3x^4+x^5-x^6)/(1-2x+x^2-x^5+2x^6-x^7)$, 1-indexed. · OeisA227582.a
a def: Determinant of the $(n+1) \times (n+1)$ Hankel-type matrix with $(i,j)$-entry equal to A005259$(i+j)$ for all $i,j = 0,\dots,n$. The entry function A005259 is taken to be $\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$. · OeisA228143.a
a def: Numbers $n$ such that $n^2 + \pi(n)$ is prime. · OeisA228828.a
a def: The multiplicative order of 2 modulo $2n+1$. · OeisA2326.a
a def: Number of parts in the symmetric representation of $\sigma(n)$. $a(n)$ is $1$ plus the number of pairs $(d_k, d_{k+1})$ of consecutive divisors of $n$ such that $d_{k+1}$ is odd and $d_{k+1} \ge 2 d_k$. · OeisA237271.a
a def: Central trinomial coefficients: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} \binom{n}{2k} \binom{2k}{k}$. · OeisA2426.a
a def: The sequence $$a(n) = \sum_{k=1}^n (-1)^{\operatorname{isprime}(k)} 10^k$$ where the sign is $-1$ if $k$ is prime, and $1$ if $k$ is not prime. · OeisA243106.a
a def: The determinant of the $n \times n$ Hankel matrix of primes. · OeisA24356.a
a def: Central factorial numbers: $a(n) = 4^n (n!)^2$. · OeisA2454.a
a def: Smallest prime factor of $2^{2^n+2} + 3$. · OeisA248802.a
a def: a: Number of partitions of $n$ into distinct parts that are not squarefree. This is the number of finite subsets of positive integers $P$ such that $\sum_{k \in P} k = n$ and every element $k \in P$ is not squarefree. · OeisA256012.a
a def: a: A total of $n$ married couples, including a mathematician M and his wife, are to be seated at the $2n$ chairs around a circular table, with no man seated next to his wife. After the ladies are seated at every other chair, M is the first … · OeisA258667.a
a def: OEIS A260194, shifted so that Lean index zero is the first OEIS term, with recurrence $a(n+3) = a(n+2) + \gcd(a(n+2),a(n))$ and initial values $a(0)=a(1)=a(2)=1$. · OeisA260194.a
a def: Decimal representation of the middle column of the "Rule 167" elementary cellular automaton starting with a single ON (black) cell. The term $a(n)$ is the decimal value of the binary number $C_{0, 0} C_{1, 0} \dots C_{n, 0}$, where $C_{i, … · OeisA267581.a
a def: Second most significant bit of the tribonacci number A000073(n). This is formalized by extracting the bit at position $\lfloor \log_2 T_n \rfloor - 1$. · OeisA271591.a
a def: a: $a(n) = \text{hypergeometric}([n, -n], [], -1)$. This is equivalent to the combinatorial sum: $$a(n) = \sum_{k=0}^n \binom{n}{k} \binom{n+k-1}{k} k!$$ The expression uses $\mathbb{N}$ arithmetic throughout, safely handling the … · OeisA278070.a
a def: Period of cubes mod $n$. The $n$-th term $a(n)$ is the smallest positive integer $T$ such that $\forall k \in \mathbb{N}$, $(k+T)^3 \equiv k^3 \pmod n$. · OeisA282779.a
a def: A028859 (OEIS): $a(n+2) = 2 \cdot a(n+1) + 2 \cdot a(n)$; $a(0) = 1$, $a(1) = 3$. · OeisA28859.a
a def: a: $\mathrm{a}(n) = \sum_{k=0}^n \mathrm{sign}(\mathrm{A007953}(5k) - \mathrm{A007953}(k))$. $\mathrm{A007953}(n)$ is the digital sum of $n$ in base 10. The sequence is non-negative, so the sum over $\mathbb{Z}$ is converted to … · OeisA289411.a
a def: The sequence $a(n)$ is defined by $a(n) = \binom{2n}{n}^3$. We use Nat.choose (2 * n) n for the central binomial coefficient. · OeisA2897.a
a def: $a(n)$ is the number of steps needed to reach a stable configuration in the 1D cellular automaton initialized with one cell with mass $n$ and based on the rule "each cell gives half of its mass, rounded down, to its right neighbor". The … · OeisA300997.a
a def: The $n$-th number $k$ such that the base $b$ expansion of $k$ for each $b = 3..k-1$ never contains more than two distinct digits. · OeisA306424.a
a def: $a(n)$ is the number of natural bases $b < 2n+1$ such that $b^n \equiv -1 \pmod{2n+1}$. The bases $b$ are interpreted as $b \in \{1, 2, \dots, 2n\}$. We check the condition in the ring $\mathbb{Z}/(2n+1)\mathbb{Z}$. · OeisA307865.a
a def: $a(n)$ is the denominator of $F(n)$ = A027641(n-1)/n + A027642(n-1)/n^2. · OeisA309132.a
a def: A binomial coefficient sum: $a(n) = \sum_{k=0}^{\lfloor n/2 \rfloor} (\binom{n}{k} - \binom{n}{k-1})^3$. · OeisA3161.a
a def: A binomial coefficient summation: $a(n) = S(3, n) / S(1, n)$. · OeisA3162.a
a def: A317940: the numerator of f n. · OeisA317940.a
a def: Coefficients of G.f. $\sum_{n\ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 + x^{n+1})^{n+1}}$. The $m$-th term $a(m)$ is the coefficient of $x^m$, which is explicitly given by the sum: $$ a(m) = \sum_{n=0}^m \sum_{k=0}^n \binom{n}{k} (-1)^j … · OeisA323557.a
a def: Coefficients of G.f. $\sum_{n \ge 0} x^n \cdot \frac{(1 + x^n)^n}{(1 - x^{n+1})^{n+1}}$. · OeisA325046.a
a def: Numerators of a sequence of fractions converging to $e$. $$a(1) = 3, a(2) = 5$$ For $n > 2$: $$a(n) = \begin{cases} \left(\frac{n+2}{2}\right) a(n-1) - a(n-2) - \left(\frac{n-2}{2}\right) a(n-3) & \text{if } n \text{ is even} \cr 2 a(n-1) … · OeisA340737.a
a def: $a(n) = \lfloor r \cdot \lfloor r \cdot n \rfloor \rfloor$, where $r = (2 + \sqrt{5})/2$. Note: The original OEIS definition has $n$ starting at 1. We define $a(n)$ for all $\mathbb{N}$. · OeisA341254.a
a def: Smallest number $k$ such that $kn + 1$ is prime. · OeisA34693.a
a def: Smallest prime $\equiv 1 \pmod n$. · OeisA34694.a
a def: A357513: $a(n) = \text{numerator of } \sum_{k = 1..n} \frac{1}{k^3} \binom{n}{k}^2 \binom{n+k}{k}^2 \text{ for } n \ge 1 \text{ with } a(0) = 0$. · OeisA357513.a
a def: A358684: $a(n)$ is the minimum integer $k$ such that the smallest prime factor of the $n$-th Fermat number exceeds $2^{2^n - k}$. Let $F_n = 2^{2^n} + 1$ be the $n$-th Fermat number, and $P_n$ be its smallest prime factor. The definition … · OeisA358684.a
a def: Denominator of the continued fraction $1/(2-3/(3-4/(4-5/(...(n-1)-n/(-2)))))$. The sequence is defined by the formula: $$a(n) = \frac{n^2 - 2}{\gcd(n^2 - 2, 2 \cdot A051403(n-3) + n \cdot A051403(n-4))}$$ The formula is valid for $n \ge 3$. · OeisA363102.a
a def: Denominator of the continued fraction $$\frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{-4}}}}}} $$ The value of the continued fraction is $C_n = 1/R_2(n)$. If $R_2(n) = N/D$ in reduced form, $C_n = … · OeisA363347.a
a def: The factorial ratio $$a(n) = \frac{(12n + 6)! \cdot (6n + 9)!}{108 \cdot (4n + 2)! \cdot (2n + 3)! \cdot ((6n + 5)!)^2}$$ It is conjectured that $a(n)$ are integers. · OeisA368692.a
a def: Denominator of the continued fraction $$ \frac{1}{2 - \frac{3}{3 - \frac{4}{4 - \frac{5}{\dots - \frac{n-1}{(n-1) - \frac{n}{n+4}}}}}} $$ · OeisA372761.a
a def: Number of primes strictly less than $n^3$. · OeisA38098.a
a def: Number of primes strictly less than $n^2$. · OeisA38107.a
a def: The sequence defined by the mutual recurrence relations: $a(n) = a(n-1)b(n-2) + a(n-2)b(n-1)$ and $b(n) = a(n-1)b(n-2) - a(n-2)b(n-1)$ starting with $a(0) = b(0) = b(1) = 1$ and $a(1) = 2$. The terms are in $\mathbb{Z}$ due to negative … · OeisA382590.a
a def: $a(n)$ is the smallest composite number $c$ such that $\textrm{primorial}(n) + c$ is prime. · OeisA38771.a
a def: The sequence of prime numbers, with $a(0) = 0$ and $a(n) = p_n$ for $n \ge 1$. · OeisA40.a
a def: The n-th partition number. · OeisA41.a
a def: Least positive multiple of $n$ using only 0's and 1's in base 10. · OeisA4290.a
a def: Denominators of coefficients in Stirling's expansion for $\log(\Gamma(z))$. · OeisA46969.a
a def: $a(n) = \sum_{k=1}^n (k^2 \bmod n)$. · OeisA48153.a
a def: Nearest integer to $n/\sqrt{2}$. · OeisA49473.a
a def: Number of nonempty subsets of $\{1, 2, 3, \dots, n\}$ whose elements have an integer average. · OeisA51293.a
a def: Maximum exponent in the prime factorization of $n$. · OeisA51903.a
a def: Apéry numbers: $a(n) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}$. · OeisA5258.a
a def: $a(n) = \sum_{k=0}^{n-1} \frac{1}{k+1} \binom{2k}{k} \binom{k}{n-1-k}$. · OeisA52709.a
a def: $a(n) = (\text{smallest prime} > n^2) - n^2$. · OeisA53000.a
a def: $a(n)$ is the concatenation of the next $n$ numbers. · OeisA53067.a
a def: Catalan-Larcombe-French sequence. · OeisA53175.a
a def: Least $m$ such that $\phi(m) = n!$. · OeisA55487.a
a def: Numerator of $1/\det(M)$ where $M$ is the $n \times n$ matrix with $M[i,j] = 1/\operatorname{lcm}(i+1,j+1)$. · OeisA60841.a
a def: Number of different products of any subset of $\{1, 2, \dots, n\}$. · OeisA60957.a
a def: First multiple of $n$ whose reverse is also divisible by $n$, or 0 if no such multiple exists. · OeisA62567.a
a def: $a(n) = \text{numerator}(H_n) - \text{denominator}(H_n)$. · OeisA64169.a
a def: Integer part of area of a regular polygon with $n$ sides each of length 1. · OeisA64313.a
a def: The subword complexity function: a(n) is the number of distinct subwords of length n. · OeisA6697.a
a def: Decimal encoding of the prime factorization of $n$. · OeisA67599.a
a def: OEIS A067836, with the convenience value a 0 = 1 preceding its official first term. · OeisA67836.a
a def: The sequence $a(n) = n! \sum_{d \mid n} \mu(n/d) H_d$ for $n \ge 1$, and $a(0) = 0$. · OeisA67857.a
a def: Number of times $n^2 + s^2$ is prime for positive integers $s < n$. · OeisA69004.a
a def: Number of primes $p$ such that $n^n \le p \le n^n + n^2$. · OeisA69922.a
a def: Number of primes $p$ such that $2^n \le p \le 2^n + \mathrm{prime}(n)$. · OeisA69923.a
a def: Catalan-Mersenne numbers: $a(0) = 2$, $a(n+1) = 2^{a(n)} - 1$. · OeisA7013.a
a def: Value of the $n$-th cyclotomic polynomial at $n$. · OeisA70518.a
a def: The sequence $a(1)=0, a(2)=1, a(n+2)=|\text{concat}(a(n+1),a(n))-\text{concat}(a(n),a(n+1))|$. · OeisA70823.a
a def: Determinant of the $n \times n$ matrix with $(i,j)$ entry $1$ if $(i+1)^2 + (j+1)^2$ is prime, and $0$ otherwise. · OeisA71524.a
a def: The sequence $a(n) = -\sum_{k=1}^n (-1)^{\lfloor (3/2)^k \rfloor}$. · OeisA71532.a
a def: Smallest $k$ such that $k!$ contains exactly $n$ 6's in base 10, or 0 if no such $k$ exists. · OeisA72200.a
a def: Wolstenholme numbers: numerator of $\sum_{k=1}^n \frac{1}{k^2}$. · OeisA7406.a
a def: Sum of the next $n$ primes, with $a(0) = 0$. · OeisA7468.a
a def: Number of times the binary pattern of $n$ occurs as an infix of the binary pattern of $n^2$. · OeisA76141.a
a def: Smallest positive integer $x$ such that $\sigma_1(x) \bmod x = n$, or $0$ if no such $x$ exists. · OeisA76495.a
a def: Trajectory of 103 under the Reverse and Add! operation in base 3. · OeisA77408.a
a def: $a(1)=1, a(2)=1, a(n)=(2^{a(n-1)} + 1)/a(n-2)$. · OeisA78590.a
a def: Smallest $m > 0$ such that $n \cdot 2^m + 1$ is prime, or 0 if no such $m$ exists. · OeisA78680.a
a def: Least positive integer $k$ such that $(k+1)(k+2)\cdots(k+n) + 1$ is prime, or 0 if no such $k$ exists. · OeisA78729.a
a def: Least prime $\ge n$ (version 1 of the "next prime" function). · OeisA7918.a
a def: $a(n) = \sum_{k=0}^n \binom{2k}{k}^3$. · OeisA79727.a
a def: Number of prime powers strictly between the $n$-th prime and the $(n+1)$-th prime. · OeisA80101.a
a def: Denominator of $\sum_{k=1}^n k^{\mu(k)}$. · OeisA80326.a
a def: Smallest positive palindrome with exactly $n$ divisors, or $0$ if no such number exists. · OeisA83753.a
a def: Smallest prime $p$ such that $p + n$ is an $n$-th power, or $0$ if no such prime exists. · OeisA84046.a
a def: Sequence $a(n)$ is the smallest $r > 0$ such that the concatenation of $n$, $r$ times with itself, multiplied by $10$ plus $1$, is prime, or $0$ if no such prime exists. · OeisA86766.a
a def: Binary representation of the prime factors of $n$, represented in decimal: $a(n) = \sum_{p \in \mathrm{support}(\mathrm{factorization}(n))} 2^{\pi(p) - 1}$. · OeisA87207.a
a def: The primary defining sequence a, satisfying $a(n) = 2 a(n-1) - 3 a(n-2)$ for $n \ge 2$, with initial values $a(0)=1$ and $a(1)=1$. · OeisA87455.a
a def: Smallest prime of the form $n, n-1, \dots, n-k$ for $k < n$, or $0$ if none exists. · OeisA87571.a
a def: The sequence a(n): least m such that countExceeding n m > countNotExceeding n m. · OeisA87719.a
a def: $a(n) = n$ if $n$ is a prime, otherwise $a(n) = 1$. · OeisA89026.a
a def: Number of pairs of twin primes $(p, p+2)$ strictly between $n^2$ and $(n+1)^2$. · OeisA91591.a
a def: $a(n) = \frac{2^{n-1}}{n!} \prod_{k=1}^{n-1} (2^k-1)$. The sequence $a(n)$ is composed of natural numbers, thus we define it as a function $\mathbb{N} \to \mathbb{N}$. · OeisA91669.a
a def: Score at stage $n$ in "tug of war" between prime gap increases vs. prime gap decreases. · OeisA92243.a
a def: Product of all composite numbers in the interval $[\binom{n}{2} + 1, \binom{n+1}{2}]$. · OeisA93456.a
a def: $a(n) = \gcd(\mathrm{numerator}(H_n), n!)$, where $H_n$ is the $n$-th harmonic number. · OeisA93818.a
a def: The Euclid-Mullin sequence, extended by a 0 = 1. · OeisA945.a
a def: Recurrence $a(n) = (a(n-1) + a(n-2)) \pmod n$. · OeisA96535.a
a' def: The "original" definition: $a'(n)$ is the minimum $k$ such that $P_n > 2^{2^n - k}$. We use Nat.find which returns the smallest natural number satisfying a predicate. · OeisA358684.a'
A001227 def: Number of odd divisors of n (A001227). · OeisA237271.A001227
A001953 def: A001953: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)\sqrt{2} \rfloor$ for $k \ge 0$. · OeisA49473.A001953
A001954 def: A001954: Nonhomogeneous Beatty sequence $\lfloor (k + 1/2)(2 + \sqrt{2}) \rfloor$ for $k \ge 0$. · OeisA49473.A001954
a005187 def: A005187: the sum of $\lfloor e / 2^k \rfloor$ over $k \ge 0$. · OeisA317940.a005187
a046644 def: A046644: the multiplicative function whose value at each prime power $p^e$ is $2^{\operatorname{A005187}(e)}$. · OeisA317940.a046644
a051403 def: Auxiliary sequence A051403, defined as $$\frac{(n+2) \sum_{k=0}^n k!}{2}$$ · OeisA363102.a051403
A239657 def: Number of odd divisors m of n such that there is a divisor d of n with d < m < 2*d (A239657). · OeisA237271.A239657
abPair def: Helper function for a, computing the pair $(a(n), b(n))$ such that: $a(n) = a(n-1)b(n-2) + a(n-2)b(n-1)$ $b(n) = a(n-1)b(n-2) - a(n-2)b(n-1)$ · OeisA382590.abPair
AbundancyIndex def: The abundancy index is the sum of the divisors of $n$ divided by $n$. · Erdos470.AbundancyIndex
ABφθSpec def: Eq. 1-4 of [Ro18], which specifies the constants $A$, $B$, $\varphi$, and $\theta$ of [Ge92]. · MovingSofa.GerversSofa.ABφθSpec
addCategoryEntry def: def addCategoryEntry {m : Type → Type} [MonadEnv m] (declName : Name) (cat : Category) (comment : String) : m Unit · ProblemAttributes.addCategoryEntry
addFormalProofAttribute def: Records the existence and location of a formal proof for a statement. · ProblemAttributes.addFormalProofAttribute
addFormalProofEntry def: def addFormalProofEntry {m : Type → Type} [MonadEnv m] (declName : Name) (kind : FormalProofKind) (link : String) (conditions : List Name · ProblemAttributes.addFormalProofEntry
additionChainLength def: The length $\ell(n)$ of $n$: the minimal number of addition steps (the number of entries minus one) over all addition chains ending at $n$. · additionChainLength
additionChainSteps def: The set of step counts realised by chains ending at n. · additionChainSteps
AdditiveBasisCondition def: Let A ⊆ ℕ be a set such that every integer can be written as n^2 + a for some a in A and n ≥ 0. · Erdos33.AdditiveBasisCondition
additiveChar def: The additive character e(x) = e ^ (2 * π * i * x), a shorthand used across several exponential-sum problems. · additiveChar
addSubjectEntry def: def addSubjectEntry {m : Type → Type} [MonadEnv m] (name : Name) (subjects : List AMS) (informal : String) : m Unit · ProblemAttributes.addSubjectEntry
Adj def: Two lattice points are adjacent iff they differ by exactly 1 in exactly one coordinate. · Erdos1212.Adj
Admissible def: P has $n$ points and at least $c n^2$ lines that each contain more than three points of P. · Erdos102.Admissible
Admissible def: A ⊆ {1, …, N} and every m has at most r representations m = p a. · Erdos538.Admissible
Admissible def: Exactly k homothetic n-cubes can tile the unit n-cube. · Erdos769.Admissible
Admissible def: The graphs the problem quantifies over: n vertices, at least $cn^2$ edges, and every edge in a triangle. · Erdos80.Admissible
Admissible def: A configuration represents a set of size $n$ (injective) and has at least two distinct distances, so that $d_2$ exists. · Erdos959.Admissible
admissibleConstants def: The set of constants $c$ such that, for all sequences of $n$ distinct positive real numbers $x_1,\ldots,x_n$, $$ \max\left(\sum x_{i_r}\right) > (c-o(1))\frac{1}{\sqrt{n}}\sum x_i $$ (where the maximum is taken over all monotonic … · Erdos1026.admissibleConstants
AdmissibleIndex def: An MZV index is admissible if it is either empty or if the first entry is at least 2 and all entries are positive. The empty list convention ensures $\mathcal{Z}_0 = \mathbb{Q}$. · ZagierMZV.AdmissibleIndex
AdmitsLieGroupStructure def: A topological group admits a Lie group structure if it has a LieGroupPresentation in some finite dimension. · AdmitsLieGroupStructure
aeval def: The evaluation of a regular function f over k at some point a with coordinates in some algebra over k · MvPolynomial.RegularFunction.aeval
aGeneral def: A general sequence defined by the Pentanacci $\pi$ recurrence, starting with arbitrary initial values $v: \text{Fin } 5 \to \mathbb{N}$. The sequence $a_{\mathrm{general}}(v, n)$ is the n-th term (0-indexed). · OeisA100478.aGeneral
AgohGiugaCongr def: The Agoh-Giuga Conjecture, Agoh's formulation. An integer p ≥ 2 is prime if and only if we have p*B_{p-1} ≡ -1 [MOD p] · AgohGiuga.AgohGiugaCongr
AgohGiugaSum def: The Agoh-Giuga Conjecture, Giuga's formulation. An integer p ≥ 2 is prime if and only if it satisfies the congruence ∑_{i=1}^{p-1} i^{p-1} ≡ -1 [MOD p]. · AgohGiuga.AgohGiugaSum
ahlswedeKhachatrian def: The Ahlswede–Khachatrian set associated to a prime factor q of N: those integers in [1, N] which are a multiple of at least one of 2 q₁, …, 2 qⱼ, q₁ ⋯ qⱼ, where q₁ < ⋯ < qⱼ = q are the prime factors of N which are at most q. · Erdos534.ahlswedeKhachatrian
aInt def: Extension of the sequence $a$ to all integers $n \in \mathbb{Z}$. For $n < 0$, $a(n) = \sum_{k=0}^{-n} \binom{n}{k}^3 \sum_{j=0}^k \binom{k}{j}^3$, where $\binom{n}{k} = (-1)^k \binom{-n+k-1}{k}$. · OeisA141057.aInt
Ak def: A101779: Ak n k is true if for all $i$ from $1$ to $n$, $i \cdot k + (i - 1)$ is prime. · OeisA101779.Ak
AllBigOmega def: All elements of A are positive and have exactly k prime factors, counted with multiplicity. · Erdos535.AllBigOmega
allEqual def: All indices equal on Fin N (using the canonical ordered vertex list). · MonochromaticQuantumGraph.allEqual
allEqualList def: Chain-equality along a list of vertices. · MonochromaticQuantumGraph.allEqualList
AllFactorialSubtractionsComposite def: def AllFactorialSubtractionsComposite (n : ℕ) : Prop · Erdos1059.AllFactorialSubtractionsComposite
AllModProdEqualsOne def: The proposition that the modular product of a collection of consecutive interval equals $1$ modulo $p$, where intervals are defined by a function specifying the consecutive boundaries. · Erdos1056.AllModProdEqualsOne
AllowedSet structure: We say a subset of $[N]^2$ is allowed for some $k$ if it contains no $k$ points which lie on a common line. · Green72.AllowedSet
AllowedSetSize def: The maximal size of an allowed set · Green72.AllowedSetSize
allUniqueSums def: allUniqueSums A is the set of elements in α that can be written as the sum of exactly one unordered pair of elements from A. · allUniqueSums
AlmostPerfect def: A number is almost perfect if the sum of its divisors is equal to $2n - 1$. · AlmostPerfectNumbers.AlmostPerfect
almostSquareRoot def: def almostSquareRoot (ε : ℝ) (N : ℕ) : ℝ · Erdos14.almostSquareRoot
alpha def: The odd-step coefficient of the Rabinowitz–Gilbert/Stoll binary recurrence. · Erdos482.alpha
Alphabet def: def Alphabet · Alphabet
alphaConst def: The constant $\alpha = \log(10)/\log(\phi) - 4$. · OeisA105565.alphaConst
alphaCore def: The $\alpha$-core of $G$: the set of vertices whose removal strictly decreases the independence number. A vertex $v$ is in the $\alpha$-core if $\alpha(G - v) < \alpha(G)$. · WrittenOnTheWallII.GraphConjecture101.alphaCore
alphaTwo def: Maximum cardinality of a 2-independent vertex set. · SimpleGraph.alphaTwo
AMS inductive: inductive AMS /-- General and overarching topics -/ · AMS
AMSLinter def: The problem category linter checks that every theorem/lemma/example has been given an AMS attribute. · AMSLinter.AMSLinter
angleSum def: $a_n = \sum_{k=1}^n \arctan(1/k)$. · Arxiv.«2607.05739».angleSum
anisotropicDims def: The set of n such that there is a nondegenerate anisotropic quadratic form on Fin n → F. Its greatest element, when it exists, is the $u$-invariant of F; the set is not bounded above exactly when the $u$-invariant of F is infinite. · QuadraticForm.anisotropicDims
annihilationNumber def: The annihilation number of a graph. This is the largest number of degrees that can be added together without going over the total number of edges of that graph. · SimpleGraph.annihilationNumber
annihilationNumber' def: Computes the annihilation number of a graph G. · SimpleGraph.annihilationNumber'
answerElab def: Indicates where the answer is in a problem statement. · Google.answerElab
AnswerInfo structure: An answer: a term, and the context in which it was elaborated · Google.AnswerInfo
answerLinter def: The actual linter object · AnswerLinter.answerLinter
AnswerSetting inductive: A type that captures the current setting for the answer() elaborator. · Google.AnswerSetting
answerSorries def: The answer(..) nodes in stx whose answer is still a sorry. · CategoryAnswerLinter.answerSorries
antidiagonalIndex def: The index $c$ of the antidiagonal containing $n$, i.e., the largest $c$ such that $T_c \le n$. This is derived from $c = \lfloor (\sqrt{8n+1} - 1) / 2 \rfloor$. · OeisA105020.antidiagonalIndex
antiRamseyNum def: The anti-Ramsey number $\mathrm{AR}(n, H)$: the maximum number of colors in an edge coloring of $K_n$ (that is, a labeling of the edges of $K_n$ using every color) that contains no rainbow copy of $H$, i.e. no injective homomorphism (copy) … · SimpleGraph.antiRamseyNum
anySizeEquilateralTriangleCount def: Number of equilateral triangles of any positive side length spanned by a finite point set. · Erdos755.anySizeEquilateralTriangleCount
ap def: Note that normally this is written as p + 1 - #E(𝔽ₚ), but since we don't have a point at infinity on this affine curve we only have p. The trace is integer-valued and can be negative. · ModularityConjecture.WeierstrassCurve.ap
aperiodicAutocorrelation def: The aperiodic autocorrelation $C_k(a)=\sum_i a_i a_{i+k}$. · Arxiv.«2104.00502».aperiodicAutocorrelation
aperyLike def: The auxiliary sequence used for the Hankel matrix, defined as $$\sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k}^2$$ · OeisA228143.aperyLike
aperyPoly def: The polynomial associated with the $n$-th Apéry number: $a_n(x) = \sum_{k=0}^n \binom{n}{k}^2 \binom{n+k}{k} x^k$. · OeisA5258.aperyPoly
Appears def: An injective vertex map carrying every edge of F to an edge of H. · UniformHypergraph.Appears
applyMove def: Apply a move by claiming x and removing it from the unclaimed pool. · Erdos872.applyMove
aprime def: The maximum cardinality among all independent sets s that maximize the quantity |s| - |N(s)|, where N(s) is the neighborhood of the set s. · SimpleGraph.aprime
aRat def: def aRat (n : ℕ) : ℚ · OeisA108.aRat
area def: The exact area of a regular $n$-gon with side length 1. · OeisA64313.area
aReal def: def aReal (n : ℕ) : ℝ · OeisA51293.aReal
arithmeticFactor def: The arithmetic factor expected to appear in the asymptotic behaviour of the $k$th moment of the Riemann zeta function, defined by $$ a_k = \prod_{p \text{ prime}} (1 - 1/p)^{k^2} \sum_{m=0}^{\infty} \binom{m+k-1}{m}^2 p^{-m}. $$ For $k \ge … · RiemannZetaMoments.arithmeticFactor
ArtinConstant def: Artin's Constant is defined to be the product $$\prod_{p\ \text{prime}} \left(1 - \frac{1}{p(p - 1)}\right)$$. · ArtinPrimitiveRootsConjecture.ArtinConstant
ASet def: The set of integers not divisible by any u_i. · Erdos1101.ASet
aStar def: The largest element of $A$ in $[1,x]$, that is $a^*(x)=\max A\cap [1,x]$ (equal to $0$ when there is no such element). · Erdos785.aStar
aStartAt def: Definition of the generalized sequence starting at x. · OeisA100475.aStartAt
aSubsequenceReal def: The sequence $b(n) = a(m*n)$ lifted to ℝ · OeisA103885.aSubsequenceReal
asymptoticSumPart def: The infinite series part of the asymptotic expansion: $\sum_{k \ge 1} \frac{(-1)^k}{k!(n-1)_k}$. · OeisA258667.asymptoticSumPart
asymptoticTerm def: The proposed asymptotic expression for a(n). · OeisA258667.asymptoticTerm
AttainableGap def: A natural number occurs as the top-two multiplicity gap of some admissible $n$-point configuration. · Erdos959.AttainableGap
augmentation def: The augmentation K^M_*(F)/ℓ → ℤ/ℓ, which sends every class {a} to 0. · MilnorK.augmentation
automorphicForms def: The automorphic forms for GL n / ℚ as a ℂ-submodule of the functions on G(𝔸). · Matrix.GeneralLinearGroup.automorphicForms
aux def: Main search loop for generating the greedy Sidon sequence. The return value for step n is the finite set of numbers generated so far, a proof that it is Sidon, and the greatest element of the finite set at that point. This is initialised … · Finset.greedySidon.aux
averageDegree def: The average degree of G. · SimpleGraph.averageDegree
averageDistance def: Average distance of G. · SimpleGraph.averageDistance
averageEccentricity def: The average eccentricity of a graph G: the mean of G.eccent v over all vertices, converted to a real number. Returns 0 if the graph has no vertices. · SimpleGraph.averageEccentricity
averageEvenDistance def: Average even distance between distinct vertices in G. Only positive even distances are considered. Returns 0 if no such distance exists. · SimpleGraph.averageEvenDistance
averageIndepNeighbors def: Average of indepNeighbors over all vertices. · SimpleGraph.averageIndepNeighbors
avoidAll def: The integers avoiding all congruences $a_i \pmod{n_i}$. · Erdos281.avoidAll
avoidPrefix def: The integers avoiding the first $k$ congruences $a_i \pmod{n_i}$. · Erdos281.avoidPrefix
avoidsPatternXYZ def: A list avoids the pattern $x \le y \le z$ if it does not have a non-decreasing subsequence of length 3. · OeisA52709.avoidsPatternXYZ
AvoidsPowersOfTwo def: A set A of natural numbers has the property in the question if a + b ≠ 2 ^ k for all a, b ∈ A (not necessarily distinct) and all k ≥ 0. · Erdos1136.AvoidsPowersOfTwo
AvoidsSubsetSum def: AvoidsSubsetSum n A means that no subset $B \subseteq A$ has sum equal to $n$. · Erdos361.AvoidsSubsetSum
A₇ def: The candidate extremal set: $\{n ∈ \{0, \dots, N-1\} : n ≡ 7 (mod 25)\}$. · Erdos848.A₇