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

1,8

Comments

First differs from A359411 at n = 128.

Links

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

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)).

Mathematica

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

Prog

(PARI) a(n) = vecprod(apply(x -> 2-hammingweight(x)%2, factor(n)[, 2]));
(Python)
from sympy import factorint
def A367516(n): return 1<<sum(1 for e in factorint(n).values() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Crossrefs

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

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

Sequence in context: A359910 A368168 A359411 * A368979 A325989 A055229

Adjacent sequences: A367513 A367514 A367515 * A367517 A367518 A367519

Keyword

nonn,easy,mult

Author

Amiram Eldar, Nov 21 2023

Status

approved