A367512 Number of evil exponents in the prime factorization of n.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0

Offset

1

Links

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

Formula

Additive with a(p^e) = A010059(e).

a(n) = A001221(n) - A293439(n).

a(n) = A001221(A367513(n)).

a(n) = log_2(A367516(n)).

a(n) >= 0, with equality if and only if n is an exponentially odious number (A270428).

a(n) <= A001221(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) = Sum_{p prime} f(1/p) = 0.12689613844142998028..., where f(x) = x - 1/2 + ((1-x)/2) * Product_{k>=0} (1-x^(2^k)).

Mathematica

f[p_, e_] := 1 - ThueMorse[e]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

(PARI) a(n) = vecsum(apply(e -> !(hammingweight(e)%2), factor(n)[, 2]));
(Python)
from sympy import factorint
def A367512(n): return sum(1 for e in factorint(n).values() if e.bit_count()&1^1) # Chai Wah Wu, Nov 23 2023

Crossrefs

Cf. A001221, A010059, A262675, A270428, A293439, A367513, A367516.

Sequence in context: A295883 A366124 A295662 * A187946 A330549 A342019

Adjacent sequences: A367509 A367510 A367511 * A367513 A367514 A367515

Keyword

nonn,easy

Author

Amiram Eldar, Nov 21 2023

Status

approved