Sequences

Quasi-Carmichael numbers to base 2: squarefree composites n such that prime p|n ==> p-2|n-2.

Quasi-Carmichael numbers to base -2: squarefree composites n such that for every prime p that divides n, p+2 divides n+2.

Quasi-Carmichael numbers to base -3: squarefree composites n such that for every prime p that divides n, p+3 divides n+3.

Quasi-Carmichael numbers to base -4: squarefree composites n such that prime p|n ==> p+4|n+4.

Quasi-Carmichael numbers to base -5: squarefree composites n such that prime p|n ==> p+5|n+5.

Quasi-Carmichael numbers to base -6: squarefree composites n such that prime p|n ==> p+6|n+6.

Quasi-Carmichael numbers to base -7: squarefree composites n such that prime p|n ==> p+7|n+7.

Quasi-Carmichael numbers to base -8: squarefree composites n such that prime p|n ==> p+8|n+8.

Quasi-Carmichael numbers to base -9: squarefree composites n such that prime p|n ==> p+9|n+9.

Quasi-Carmichael numbers to base -10: squarefree composites n such that prime p|n ==> p+10|n+10.

Number of permutations of an n-set containing a 4-cycle.

Number of permutations of an n-set containing a 5-cycle.

Number of permutations of an n-set containing a 6-cycle.

Number of permutations of an n-set containing a 7-cycle.

Number of permutations of an n-set containing an 8-cycle.

Number of permutations of an n-set containing a 9-cycle.

Number of permutations of an n-set containing a 10-cycle.

The natural numbers interleaved with the even numbers.

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

a(n+1) = [ A*a(n)+B ]/p^r, where p^r is the highest power of p diving [ A*a(n)+B ] and p=2, A=2.00013, B=3.0.

Numbers in which all digits are composite.

E.g.f. sin(x) + cos(x) + tan(x).

Expansion of sin x + cos x + tan x + sec x.

Expansion of cos x + tan x + sec x.

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

Theta series of 6-dimensional 11-modular even lattice of minimal norm 4.

a(n) = A029571(n) / 6.

a(n) = A029572(n) / 24.

Expansion of (1+sin(x)+sin(x)^2)/(1-sin(x)+sin(x)^2).

For n>0, a(n) is the least quasi-Carmichael number to base n; a(0) = least composite squarefree integer.

For n>0, a(n) is the least quasi-Carmichael number to base -n, extended to n=0 with the least composite squarefree integer.

Theta series of 6-dimensional 11-modular odd lattice of minimal norm 4.

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 3, 1; 0, 1, 4 ].

Theta series of quadratic form with Gram matrix [ 1, 0, 0; 0, 2, 0; 0, 0, 3 ].

Theta series of quadratic form (or lattice) with Gram matrix [ 1, 0, 0; 0, 2, 1; 0, 1, 3 ].

Numbers represented by quadratic form with Gram matrix [ 4, 2, 2; 2, 4, 1; 2, 1, 6 ], divided by 2.

Numbers represented by quadratic form with Gram matrix [ 3, 1, 1; 1, 2, 1; 1, 1, 3 ].

Numbers represented by quadratic form with Gram matrix [ 2, 1, 0; 1, 2, 1; 0, 1, 3 ].

Theta series of quadratic form with Gram matrix [ 4, 1, 0; 1, 2, 1; 0, 1, 6 ].

Numbers in the (2,3)-Pascal triangle (by row).

Distinct numbers in the (2,3)-Pascal triangle A029600.

Numbers in the (2,3)-Pascal triangle A029600 that are different from 2.

Numbers in the (2,3)-Pascal triangle A029600 that are different from 3.

Odd numbers in the (2,3)-Pascal triangle A029600.

Even numbers in the (2,3)-Pascal triangle A029600.

Odd numbers in the (2,3)-Pascal triangle A029600 that are different from 3.

Even numbers in the (2,3)-Pascal triangle A029600 that are different from 2.

Distinct odd numbers in (2,3)-Pascal triangle A029600.

Central numbers in the (2,3)-Pascal triangle A029600.

Numbers to the left of the central numbers of the (2,3)-Pascal triangle A029600.