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A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.

%I A372504 #10 May 04 2024 04:28:47
%S A372504 1,1,1,2,1,1,1,0,2,1,1,2,1,1,1,4,1,2,1,2,1,1,1,0,2,1,0,2,1,1,1,0,1,1,
%T A372504 1,4,1,1,1,0,1,1,1,2,2,1,1,4,2,2,1,2,1,0,1,0,1,1,1,2,1,1,2,0,1,1,1,2,
%U A372504 1,1,1,0,1,1,2,2,1,1,1,4,4,1,1,2,1,1,1
%N A372504 Multiplicative with a(p^e) = e if e is a power of 2, and 0 otherwise.
%H A372504 Amiram Eldar, <a href="/A372504/b372504.txt">Table of n, a(n) for n = 1..10000</a>
%F A372504 Multiplicative with a(p^e) = A048298(e) = A209229(e) * e.
%F A372504 a(n) = A355823(n) * A005361(n).
%F A372504 a(A138302(n)) = A005361(A138302(n)) = A368473(n).
%F A372504 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.31285540951965780409..., where f(x) = (1-x) * (1 + Sum_{k>=0} 2^k * x^(2^k)).
%t A372504 f[p_, e_] := If[e == 2^IntegerExponent[e, 2], e, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o A372504 (PARI) a(n) = vecprod(apply(x -> if(x == 1 << valuation(x, 2), x, 0), factor(n)[, 2]));
%Y A372504 Cf. A005361, A048298, A138302, A209229, A355823, A368473, A372470.
%K A372504 nonn,easy,mult
%O A372504 1,4
%A A372504 _Amiram Eldar_, May 04 2024