A125096 Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.

1, 0, 2, 2, 0, 0, 0, 2, 3, 0, 2, 4, 0, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 2, 4, 0, 0, 6, 0, 0, 0, 0, 2, 0, 2, 4, 0, 0, 0, 4, 1, 0, 4, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 4, 0, 0, 0, 6, 2, 0, 2, 4, 0, 0, 0, 0, 5, 0, 2, 0, 0, 0, 0, 4, 2, 0, 0, 0, 0, 0, 0, 4, 2, 0, 6, 2, 0, 0, 0, 0, 0

Offset: 1

Author

Michael Somos, Nov 20 2006

Keywords

• nonn
• mult

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A002324, A002325, A033761, A093954, A112603.

Programs

Mathematica

f[p_, e_] := If[MemberQ[{1, 3}, Mod[p, 8]], e + 1, (1 + (-1)^e)/2]; f[2, e_] := If[e > 1, 2, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 13 2022 *)

PARI

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])}

Formula

a(n) is multiplicative with a(2) = 0, a(2^e) = 2 if e>1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8).

a(4*n + 2) = a(8*n + 5) = a(8*n + 7) = 0. a(4*n) = 2 * A002325(n). a(8*n + 1) = A112603(n). a(8*n + 3) = A033761(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - Amiram Eldar, Oct 13 2022