

A035267


Indices of nonzero terms in expansion of Dirichlet series Product_p (1(Kronecker(m,p)+1)*p^(s)+Kronecker(m,p)*p^(2s))^(1) for m= 37.


1



1, 3, 4, 7, 9, 11, 12, 16, 21, 25, 27, 28, 33, 36, 37, 41, 44, 47, 48, 49, 53, 63, 64, 67, 71, 73, 75, 77, 81, 83, 84, 99, 100, 101, 107, 108, 111, 112, 121, 123, 127, 132, 137, 139, 141, 144, 147, 148, 149, 151, 157, 159, 164, 169, 173, 175, 176, 181, 188
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OFFSET

1,2


COMMENTS

Positive numbers represented by the indefinite quadratic form 3x^2+xy3y^2, of discriminant 37.  N. J. A. Sloane, Jun 05 2014 [Typo corrected by Klaus Purath, Apr 24 2023]
Also positive numbers of the form x^2 + (2m+1)xy + (m^2+m9)y^2, m, x, y integers. All squares as well as the products of any terms belong to the sequence. Thus, this set of terms is closed under multiplication.  Klaus Purath, Apr 24 2023
A positive integer k belongs to the sequence if and only if k (modulo 37) is a term of A010398 and, moreover, in the case that prime factors p of k are terms of A038914, they occur only with even exponents. Or, more briefly, any positive integer is a term of this sequence if none of its divisors is an odd power of primes from A038914. For these primes also p (modulo 37) = {2, 5, 6, 8, 13, ...} = A028750 applies.  Klaus Purath, May 12 2023


LINKS



MATHEMATICA

Reap[For[n = 0, n <= 100, n++, If[ Reduce[ 3*x^2 + x*y  3*y^2 == n, {x, y}, Integers] =!= False, Sow[n]]]][[2, 1]] (* N. J. A. Sloane, Jun 05 2014 *)


PROG

(PARI) m=37; select(x > x, direuler(p=2, 101, 1/(1(kronecker(m, p)*(XX^2))X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020


CROSSREFS



KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



