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URL: https://oeis.org/A258328

⇱ A258328 - OEIS

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A258328
L.g.f.: log(1 + Sum_{n>=1} x^(n^2) + x^(3*n^2) ).
2
1, -1, 4, -1, 1, -4, 1, -1, 13, -11, 12, -16, 14, -15, 19, -1, 1, -13, 1, -11, 25, -12, 24, -40, 26, -14, 40, -15, 1, -29, 1, -1, 48, -35, 36, -61, 38, -39, 56, -11, 1, -39, 1, -12, 73, -24, 48, -88, 50, -36, 55, -14, 1, -40, 12, -15, 61, -59, 60, -101, 62, -63, 97, -1, 14, -48, 1, -35, 96, -60, 72, -157, 74, -38, 119, -39, 12, -56, 1, -11, 121, -83, 84, -135, 86, -87, 91, -12, 1, -83, 14, -24, 97, -48, 96, -184, 98, -64, 156, -36, 1, -89, 1, -14, 180, -107, 108, -196, 110, -132, 152, -15, 1, -99, 24, -59, 182, -60, 120, -245, 133
OFFSET
1,3
LINKS
FORMULA
a(n) = -1 iff n = 2^k for k>=1 [conjecture].
a(p) = +1 for primes p such that 3 is not a square mod p (A003630), and a(n) = +1 nowhere else except at n=0 [conjecture].
EXAMPLE
L.g.f.: L(x) = x - x^2/2 + 4*x^3/3 - x^4/4 + x^5/5 - 4*x^6/6 + x^7/7 - x^8/8 + 13*x^9/9 - 11*x^10/10 + 12*x^11/11 - 16*x^12/12 + 14*x^13/13 - 15*x^14/14 + 19*x^15/15 - x^16/16 +...+ a(n)*x^n/n +...
where
exp(L(x)) = 1 + x + x^3 + x^4 + x^9 + x^12 + x^16 + x^25 + x^27 + x^36 + x^48 + x^49 + x^64 + x^75 + x^81 + x^100 + x^108 +...+ x^(n^2) + x^(3*n^2) +...
Note that for n>1, a(n) = +1 at positions:
[5, 7, 17, 19, 29, 31, 41, 43, 53, 67, 79, 89, 101, 103, 113, 127, ...];
which appears to be A003630 (primes p such that 3 is not a square mod p).
PROG
(PARI) {a(n) = local(L=x); L = log(1 + sum(k=1, sqrtint(n+1), x^(k^2) + x^(3*k^2)) +x*O(x^n)); n*polcoeff(L, n)}
for(n=1, 121, print1(a(n), ", "))
CROSSREFS
KEYWORD
sign
AUTHOR
Paul D. Hanna, Jun 03 2015
STATUS
approved