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%I #12 Nov 07 2022 02:10:42
%S 1,1,1,2,1,1,1,2,2,1,1,2,1,1,1,2,1,2,1,2,1,1,1,2,2,1,2,2,1,1,1,2,1,1,
%T 1,4,1,1,1,2,1,1,1,2,2,1,1,2,2,2,1,2,1,2,1,2,1,1,1,2,1,1,2,4,1,1,1,2,
%U 1,1,1,4,1,1,2,2,1,1,1,2,2,1,1,2,1,1,1
%N a(n) is the number of infinitary square divisors of n.
%C First differs from A007424 at n = 36, from A323308 at n = 64, and from A278908 and A307848 at n = 128.
%H Amiram Eldar, <a href="/A358260/b358260.txt">Table of n, a(n) for n = 1..10000</a>
%F Multiplicative with a(p^e) = 2^A000120(e) if e is even, and 2^A000120(e-1) if e is odd.
%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} ((1-1/p) * Sum_{k>=1} a(p^k)/p^k) = 1.55454884667440993654... .
%t f[p_, e_] := 2^DigitCount[If[OddQ[e], e - 1, e], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
%o (PARI) a(n) = {my(f = factor(n)); prod(i=1, #f~, 2^hammingweight(if(f[i,2]%2, f[i,2]-1, f[i,2])))};
%Y Cf. A000120, A048881, A037445, A077609, A358261.
%Y Similar sequences: A046951, A056624, A056626.
%Y Sequences with the same initial terms: A007424, A278908, A307848, A323308.
%K nonn,mult
%O 1,4
%A _Amiram Eldar_, Nov 06 2022
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