Sequences

Number of groups of order n.

Kolakoski sequence: a(n) is length of n-th run; a(1) = 1; sequence consists just of 1's and 2's.

Number of classes of primitive positive definite binary quadratic forms of discriminant D = -4n; or equivalently the class number of the quadratic order of discriminant D = -4n.

The zero sequence.

d(n) (also called tau(n) or sigma_0(n)), the number of divisors of n.

Integer part of square root of n-th prime.

The characteristic function of {0}: a(n) = 0^n.

Number of ways of making change for n cents using coins of 1, 2, 5, 10 cents.

Expansion of Product_{m >= 1} (1 + x^m); number of partitions of n into distinct parts; number of partitions of n into odd parts.

Euler totient function phi(n): count numbers <= n and prime to n.

Number of n-bead necklaces (turning over is allowed) where complements are equivalent.

The simplest sequence of positive numbers: the all 1's sequence.

Definition (1): Number of n-bead binary necklaces with beads of 2 colors where the colors may be swapped but turning over is not allowed.

Number of series-reduced trees with n nodes.

Smallest prime power >= n.

a(n) is the number of distinct (infinite) output sequences from binary n-stage shift register which feeds back the complement of the last stage.

Erroneous version of A032522.

Number of positive integers <= 2^n of form x^2 + 16*y^2.

Number of primitive permutation groups of degree n.

Number of primitive polynomials of degree n over GF(2) (version 2).

Number of positive integers <= 2^n of form x^2 + 12 y^2.

Number of centered hydrocarbons with n atoms.

Expansion of e.g.f. exp(-2*x)/(1-x).

Number of positive integers <= 2^n of form x^2 + 10 y^2.

Coefficients of the 3rd-order mock theta function f(q).

Mosaic numbers or multiplicative projection of n: if n = Product (p_j^k_j) then a(n) = Product (p_j * k_j).

The positive integers. Also called the natural numbers, the whole numbers or the counting numbers, but these terms are ambiguous.

Let k = p_1^e_1 p_2^e_2 p_3^e_3 ... be the prime factorization of n. Sequence gives k such that the sum of the numbers of 1's in the binary expansions of e_1, e_2, e_3, ... is odd.

Number of necklaces with n beads of 2 colors, allowing turning over (these are also called bracelets).

Initial digit of n.

Number of n-bead necklaces with 2 colors when turning over is not allowed; also number of output sequences from a simple n-stage cycling shift register; also number of binary irreducible polynomials whose degree divides n.

Coefficients of ménage hit polynomials.

Period 2: repeat [1, 2]; a(n) = 1 + (n mod 2).

Period 2: repeat [0, 1]; a(n) = n mod 2; parity of n.

Let A(n) = #{(i,j): i^2 + j^2 <= n}, V(n) = Pi*n, P(n) = A(n) - V(n); A000099 gives values of n where |P(n)| sets a new record; sequence gives closest integer to P(A000099(n)).

Numbers that are not squares (or, the nonsquares).

Twice A000007.

Coefficient of q^(2n) in the series expansion of Ramanujan's mock theta function f(q).

a(n) is the number of partitions of n (the partition numbers).

Unary representation of natural numbers.

Mersenne exponents: primes p such that 2^p - 1 is prime. Then 2^p - 1 is called a Mersenne prime.

Dying rabbits: a(0) = 1; for 1 <= n <= 12, a(n) = Fibonacci(n); for n >= 13, a(n) = a(n-1) + a(n-2) - a(n-13).

Number of primitive n-bead necklaces (turning over is allowed) where complements are equivalent.

Number of integers <= 2^n of form x^2 - 2y^2.

Number of n-bead necklaces with beads of 2 colors and primitive period n, when turning over is not allowed but the two colors can be interchanged.

Number of positive integers <= 2^n of the form 3*x^2 + 4*y^2.

Number of positive integers <= 2^n of form x^2 + y^2.

a(n) = 2^n + 1.

1-digit numbers arranged in alphabetical order, then the 2-digit numbers arranged in alphabetical order, then the 3-digit numbers, etc.

Local stops on New York City 1 Train (Broadway-7 Avenue Local) subway.