A267137 Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.

0, 1, 2, 4, 5, 6, 8, 9, 10, 12, 14, 16, 17, 20, 21, 22, 24, 25, 26, 30, 32, 33, 34, 36, 37, 40, 41, 42, 44, 46, 49, 50, 52, 54, 56, 57, 58, 60, 64, 65, 66, 69, 70, 72, 74, 76, 80, 81, 82, 85, 86, 89, 90, 92, 94, 96, 97, 100, 101, 102, 104, 105, 108, 110, 112, 114, 116

Offset

1,3

Comments

Inspired by relation between A003136 and A202822. See comment section of A202822.

Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...

Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...

Obviously, A000290, A002378 and A045944 are subsequences.

The complement of this sequence is A322430. - Kemoneilwe Thabo Moseki, Dec 12 2019

Links

Table of n, a(n) for n=1..67.

Formula

a(n) = (A202822(n) - 1) / 3.

Example

1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.

Prog

(PARI) x='x+O('x^500); p=eta(x)^3/eta(x^3); for(n=0, 499, if(polcoeff(p, n) != 0 && n%3==1, print1((n-1)/3, ", ")));
(PARI) is(n) = sumdiv( n, d, kronecker( -3, d));
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));
(PARI) is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));

Crossrefs

Cf. A000290, A002378, A003136, A045944, A202822.

Sequence in context: A091529 A184967 A141820 * A095775 A035063 A004128

Adjacent sequences: A267134 A267135 A267136 * A267138 A267139 A267140

Keyword

nonn

Author

Altug Alkan, Jan 10 2016

Status

approved