Sequences

The square sieve.

Numbers k such that k and k+1 have same sum of divisors.

Number of simple imperfect squared squares of order n up to symmetry.

Number of chisel strokes required for Roman numerals for n.

Smallest number requiring n chisel strokes for its representation in Roman numerals.

Interleave denominators (A000129) and numerators (A001333) of convergents to sqrt(2).

Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n where 0 < x_1 <= ... <= x_n.

Egyptian fractions: number of solutions of 1 = 1/x_1 + ... + 1/x_n in positive integers.

Number of pairings {(b(1), c(1)), (b(2), c(2)), ..., (b(n), c(n))} of the first 2n positive integers satisfying b(i) < c(i) and such that the 2n numbers c(i)+b(i) and c(i)-b(i) are all distinct.

E.g.f. 1/(1 - sin(x) + sin(x)^2).

Numbers k such that 4*k^2 + 9 is prime.

Numbers k such that 4*k^2 + 25 is prime.

a(n) is the odd member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

a(n) is half of the even member of {x,y}, where x^2 + y^2 is the n-th prime of the form 4i+1.

Number of restricted solid partitions of n.

Primitive weird numbers: weird numbers with no proper weird divisors.

Number of certain self-avoiding walks with n steps on square lattice (see reference for precise definition).

Klarner-Rado sequence: a(1) = 1; subsequent terms are defined by the rule that if m is present so are 2m+1 and 3m+1.

Low-temperature series in y = exp(2J/kT) for antiferromagnetic susceptibility for the Ising model on honeycomb structure.

Low-temperature series in exp(4J/kT) for antiferromagnetic susceptibility for the Ising model on square lattice.

E.g.f: 1/(1 - sin(x) + sin(x)^2 - sin(x)^3).

Numbers k such that k! + 1 is prime.

Numbers k such that k! - 1 is prime.

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

a(0) = 1; for n > 0, a(n) = a(n-1) + floor(sqrt(a(n-1))).

Number of trees in an n-node wheel.

Number of non-cyclic hydrocarbons with n carbon atoms (excluding stereoisomers).

a(1) = 1; a(2) = 2; a(n) == a(k) (mod n-k) for all 1 < k < n.

Number of trimmed trees with n nodes.

Number of n-node trees with a forbidden limb of length 3.

Number of n-node trees with a forbidden limb of length 4.

Number of n-node trees with a forbidden limb of length 5.

Number of n-node trees with a forbidden limb of length 6.

Initial digits of squares.

Initial digit of cubes.

Number of unlabeled planar trees (also called plane trees) with n nodes.

a(n) = Sum_{k|n} mu(k)*Catalan(n/k) (mu = Moebius function A008683).

Carmichael numbers: composite numbers k such that a^(k-1) == 1 (mod k) for every a coprime to k.

Smallest multiple of n whose digits sum to n.

Expansion of (1 + x*exp(x))^2.

Number of bifix-free (or primary, or unbordered) words of length n over a two-letter alphabet.

Smallest number of multiplicative persistence n.

Size of the largest subset of the numbers [1...n] which does not contain a 3-term arithmetic progression.

Size of the largest subset of the numbers [1...n] which doesn't contain a 4-term arithmetic progression.

Size of the largest subset of the numbers [1..n] which does not contain a 5-term arithmetic progression.

Size of the largest subset of the numbers [1..n] which doesn't contain a 6-term arithmetic progression.

Number of n-level ladder expressions with A001622.

Erroneous version of A082499.

Number of n-level ladder expressions with A030798.

a(n+1) = a(n) + 4^(n-1) a(n)^2.