Sequences

a(n) = a(n-1)*a(n-2) + 1 with a(0) = a(1) = 0.

Triple factorial numbers a(n) = n!!!, defined by a(n) = n*a(n-3), a(0) = a(1) = 1, a(2) = 2. Sometimes written n!3.

Quadruple factorial numbers n!!!!: a(n) = n*a(n-4).

Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).

Reve's puzzle: number of moves needed to solve the Towers of Hanoi puzzle with 4 pegs and n disks, according to the Frame-Stewart algorithm.

Tower of Hanoi with 5 pegs.

a(n) = smallest number k such that k^n is the sum of n positive n-th powers, or 0 if no solution exists.

The sum of both two and three consecutive squares.

Numbers k such that phi(k) divides sigma(k) and sigma(k)/k > sigma(m)/m for all m < k.

Duplicate of A034343.

Numbers n such that 2^n - 2^((n + 1)/2) + 1 is prime.

Numbers n such that 2^n + 2^((n + 1)/2) + 1 is prime.

a(n) = A002034(n)!/n.

Number of coins needed for ApSimon's mints problem.

Numbers m such that m and m+1 are squarefree.

Numbers m such that m, m+1 and m+2 are squarefree.

Numerators of convergents to e.

Denominators of convergents to e.

Number of regions in regular n-gon with all diagonals drawn.

If n mod 4 = 0 then 2^(n-1)+1 elif n mod 4 = 2 then 2^(n-1)-1 else 2^(n-1).

a(n) = (2n+1)*n!.

a(n) = (2*n+1)^2*n!.

a(n) = -Sum_{k = 0..n-1} (n+k)!a(k)/(2k)!.

a(1) = 1; a(n) = -Sum_{k = 1..n-1} (n+k)!a(k)/(2k)!.

Prime(n)*...*prime(a(n)) is the least product of consecutive primes that is non-deficient.

a(n) = Product_{k=1..n} binomial(2*k,k).

Prime(n)*...*a(n) is the least product of consecutive primes which is non-deficient.

Number of 4-colorings of cyclic group of order n.

Number of 5-colorings of cyclic group of order n.

a(n) = 2^n + 3^n.

Number of partitions of n in which no part occurs just once.

Multiply-perfect numbers: n divides sigma(n).

Numbers that are the sum of 2 nonzero squares in 2 or more ways.

Primes p such that 6*p + 1 is also prime.

Numbers k such that phi(k) divides k.

Cardinalities of Sperner families on 1,...,n.

Quartic (or 4-fold) factorial numbers: a(n) = Product_{k = 0..n-1} (4*k + 1).

Smallest odd number expressible in at least n ways as p+2*m^2 where p is 1 or a prime and m >= 0.

a(n) = 22*a(n-1) - 3*a(n-2) + 18*a(n-3) - 11*a(n-4). Deviates from A007699 at the 1403rd term.

Pisot sequence E(10,219): a(n) = nearest integer to a(n-1)^2 / a(n-2), starting 10, 219, ... Deviates from A007698 at 1403rd term.

Numbers n such that n, 2n+1, and 4n+3 all prime.

a(0) = 0; for n > 0, a(n) = n^n*2^((n-1)^2).

a(n) = prime(n)*...*prime(m), the least product of consecutive primes which is non-deficient.

Regular primes.

a(n+2) = (a(n) - 1)*a(n+1) + 1, with (a(1), a(2)) = (2, 3).

Number of ways of arranging 2n+1 nonattacking queens on a 2n+1 X 2n+1 toroidal board.

a(n) = 1 + coefficient of x^n in Product_{k>=1} (1-x^k) (essentially the expansion of the Dedekind function eta(x)).

Prime(n)*...*prime(a(n)) is the least product of consecutive primes which is abundant.

Prime(n)*...*a(n) is the least product of consecutive primes which is abundant.

Number of winning (or reformed) decks at Mousetrap.