OFFSET
0,2
FORMULA
If n>0 then a(n) = -3 * b(n) where b is multiplicative with b(2^e) = 2 - 2^e, b(5^e) = 1, and b(p^e) = (p^(e+1) - 1) / (p - 1) otherwise.
G.f.: 1 - 3 * Sum_{k>0} c(k) * x^k / (1 - x^k)^2 where c(k) is a period 10 integer sequence.
G.f.: 1 - 3/2 * Sum_{k>0} c(k) * k * x^k / (1 - x^k) where c(k) is a period 10 integer sequence.
a(4*n) = A028887(n). a(4*n + 2) = 0.
EXAMPLE
G.f. = 1 - 3*q - 12*q^3 + 6*q^4 - 3*q^5 - 24*q^7 + 18*q^8 - 39*q^9 + ...
PROG
(PARI) {a(n) = local(A, p, e); if( n<1, n==0, A = factor(n); -3 * prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 2 - 2^e, if( p==5, 1, (p^(e+1) - 1) / (p - 1))))))};
(PARI) {a(n) = if( n<1, n==0, -3 * sumdiv(n, k, n/k * [8, 1, -2, 1, -2, -4, -2, 1, -2, 1][k%10 + 1]))};
(PARI) {a(n) = if( n<1, n==0, -3/2 * sumdiv(n, k, k * [0, 2, -1, 2, -1, 0, -1, 2, -1, 2][k%10 + 1]))};
(Magma) A := Basis( ModularForms( Gamma0(10), 2), 60); A[1] - 3*A[2];
CROSSREFS
KEYWORD
sign
AUTHOR
Michael Somos, Oct 18 2014
STATUS
approved