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The number of exponentially evil divisors of n.
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%I #10 Feb 24 2024 16:16:10

%S 1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,2,1,1,1,1,3,1,1,

%T 1,1,1,1,1,2,1,1,1,1,1,1,1,2,1,1,1,1,1,2,1,2,1,1,1,1,1,1,1,4,1,1,1,1,

%U 1,1,1,2,1,1,1,1,1,1,1,2,2,1,1,1,1,1,1

%N The number of exponentially evil divisors of n.

%C First differs from A050361 at n = 128.

%C The number of divisors of n that are exponentially evil numbers (A262675), i.e., numbers having only evil (A001969) exponents in their canonical prime factorization.

%C The sum of these divisors is A366904(n) and the largest of them is A366906(n).

%H Amiram Eldar, <a href="/A366902/b366902.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(p^e) = A159481(e).

%F a(n) >= 1, with equality if and only if n is a cubefree number (A004709).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} Sum_{k>=1} 1/p^A262675(k) = 1.241359937856... .

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

%o (PARI) s(n) = n\2 + (n%2 || hammingweight(n+1)%2); \\ after _Charles R Greathouse IV_ at A159481

%o a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

%Y Cf. A001969, A004709, A050361, A159481, A262675, A366904, A366906.

%Y Similar sequences: A325837, A353898, A365680, A366901.

%K nonn,easy,mult

%O 1,8

%A _Amiram Eldar_, Oct 27 2023