OFFSET
1,3
LINKS
Paul D. Hanna, Table of n, a(n) for n = 1..1024
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