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%I #15 Oct 13 2022 06:38:38
%S 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,
%T 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,
%U 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
%N Expansion of -1 + (phi(q) * phi(q^2) + phi(-q^2) * phi(q^4)) / 2 in powers of q.
%H Amiram Eldar, <a href="/A125096/b125096.txt">Table of n, a(n) for n = 1..10000</a>
%F 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).
%F 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).
%F 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
%t 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 *)
%o (PARI) {a(n) = if( n<1, 0, qfrep([1, 0; 0, 8], n)[n] + qfrep([3, 1; 1, 3], n)[n])}
%Y Cf. A002324, A002325, A033761, A093954, A112603.
%K nonn,mult
%O 1,3
%A _Michael Somos_, Nov 20 2006
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