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%I #36 Dec 14 2019 08:15:04
%S 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,
%T 40,41,42,44,46,49,50,52,54,56,57,58,60,64,65,66,69,70,72,74,76,80,81,
%U 82,85,86,89,90,92,94,96,97,100,101,102,104,105,108,110,112,114,116
%N Numbers of the form x^2 + x + x*y + y + y^2 where x and y are integers.
%C Inspired by relation between A003136 and A202822. See comment section of A202822.
%C Prime terms of this sequence are 2, 5, 17, 37, 41, 89, 97, 101, 137, 149, ...
%C Perfect power terms of this sequence are 1, 4, 8, 9, 16, 25, 32, 36, 49, 64, 81, 100, 121, 144, 169, ...
%C Obviously, A000290, A002378 and A045944 are subsequences.
%C The complement of this sequence is A322430. - _Kemoneilwe Thabo Moseki_, Dec 12 2019
%F a(n) = (A202822(n) - 1) / 3.
%e 1 is a term because (-1)^2 + (-1) + (-1)*(-1) + (-1) + (-1)^2 = 1.
%e 4 is a term because 2^2 + 2 + 2*(-2) + (-2) + (-2)^2 = 4.
%e 24 is a term because 2^2 + 2 + 2*3 + 3 + 3^2 = 24.
%o (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, ", ")));
%o (PARI) is(n) = sumdiv( n, d, kronecker( -3, d));
%o for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));
%o (PARI) is(n) = #bnfisintnorm(bnfinit(z^2+z+1), n);
%o for(n=0, 1e3, if(is(3*n+1), print1(n, ", ")));
%Y Cf. A000290, A002378, A003136, A045944, A202822.
%K nonn
%O 1,3
%A _Altug Alkan_, Jan 10 2016
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