A367516 The number of unitary divisors of n that are exponentially evil numbers (A262675).

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

Offset: 1

Author

Amiram Eldar, Nov 21 2023

Keywords

• nonn
• easy
• mult

Comments

First differs from A359411 at n = 128.

Links

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

Crossrefs

Cf. A010060, A262675, A270428, A359411, A367512.

Similar sequences: A034444, A055076, A056624, A366901, A366902, A367515.

Programs

Mathematica

f[p_, e_] := If[EvenQ[DigitCount[e, 2, 1]], 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]

PARI

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

Python

from sympy import factorint
def A367516(n): return 1<Chai Wah Wu, Nov 23 2023

Formula

Multiplicative with a(p^e) = (2-A010060(e)).

a(n) = A034444(n)/A367515(n).

a(n) = 2^A367512(n).

a(n) >= 1, with equality if and only if n is an exponentially odious number (A270428).

a(n) <= A034444(n), with equality if and only if n is an exponentially evil number (A262675).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Product_{p prime} f(1/p) = 1.13071730542774788785..., where f(x) = 1/2 + x + ((1-x)/2) * Product_{k>=0} (1 - x^(2^k)).