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%I #22 Aug 28 2023 08:25:35
%S 1,2,-1,3,-1,-2,-1,4,0,-2,-1,-3,-1,-2,1,5,-1,0,-1,-3,1,-2,-1,-4,0,-2,
%T 0,-3,-1,2,-1,6,1,-2,1,0,-1,-2,1,-4,-1,2,-1,-3,0,-2,-1,-5,0,0,1,-3,-1,
%U 0,1,-4,1,-2,-1,3,-1,-2,0,7,1,2,-1,-3,1,2,-1,0,-1
%N Moebius transform of A129502.
%C Dirichlet convolution of A209635 and A209229.
%H Andrew Howroyd, <a href="/A317673/b317673.txt">Table of n, a(n) for n = 1..1024</a>
%F Multiplicative with a(2^e) = e+1, and if p is an odd prime, a(p) = -1 and a(p^e) = 0 for e >= 2. - _Amiram Eldar_, Aug 28 2023
%t a[n_] := Module[{e}, Sum[e = IntegerExponent[d, 2]; If[d == 2^e, MoebiusMu[n/d] Binomial[2 + e, 2], 0], {d, Divisors[n]}]];
%t a /@ Range[1, 100] (* _Jean-François Alcover_, Sep 24 2019, from PARI *)
%t f[p_, e_] := If[e == 1, -1, 0]; f[2, e_] := e+1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 70] (* _Amiram Eldar_, Aug 28 2023 *)
%o (PARI) a(n)={sumdiv(n, d, my(e=valuation(d, 2)); if(d==1<<e, moebius(n/d) * binomial(2+e,2), 0))}
%Y First column of A129501.
%Y Cf. A129502, A209229, A209635 (Moebius transform of A104117).
%K sign,easy,mult
%O 1,2
%A _Andrew Howroyd_, Aug 03 2018
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