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The number of infinitary divisors of n that are exponentially odd numbers (A268335).
6

%I #11 Oct 21 2023 01:50:04

%S 1,2,2,1,2,4,2,3,1,4,2,2,2,4,4,1,2,2,2,2,4,4,2,6,1,4,3,2,2,8,2,3,4,4,

%T 4,1,2,4,4,6,2,8,2,2,2,4,2,2,1,2,4,2,2,6,4,6,4,4,2,4,2,4,2,1,4,8,2,2,

%U 4,8,2,3,2,4,2,2,4,8,2,2,1,4,2,4,4,4,4

%N The number of infinitary divisors of n that are exponentially odd numbers (A268335).

%C First differs from A295878 at n = 32.

%H Amiram Eldar, <a href="/A363825/b363825.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = 1 if e is even, and a(p^e) = 2^A000120(e-1) + 1 if e is odd.

%F a(n) <= A000005(n) with equality if and only if n is squarefree (A005117).

%t f[p_, e_] := 1 + If[OddQ[e], 2^DigitCount[e-1, 2, 1], 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, 2^hammingweight(f[i, 2]-1) + 1, 1))};

%Y Cf. A000005, A000120, A005117, A037445, A077609, A268335, A295878, A358260.

%K nonn,easy,mult

%O 1,2

%A _Amiram Eldar_, Oct 19 2023