A366902 The number of exponentially evil divisors of n.

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, 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, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Author

Amiram Eldar, Oct 27 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A050361 at n = 128.

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.

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

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

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

Similar sequences: A325837, A353898, A365680, A366901.

Programs

Mathematica

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]

PARI

s(n) = n\2 + (n%2 || hammingweight(n+1)%2); \\ after Charles R Greathouse IV at A159481
a(n) = vecprod(apply(x -> s(x), factor(n)[, 2]));

Formula

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

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

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... .