A350386 a(n) is the sum of the even exponents in the prime factorization of n.

0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 4, 2, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 6, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 2, 2, 0, 0, 0, 4, 4, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 2, 2, 4, 0, 0, 0, 0, 0

Offset: 1

Author

Amiram Eldar, Dec 28 2021

Keywords

• nonn
• easy

Comments

a(n) is the number of prime divisors of n, counted with multiplicity, with an even exponent in the prime factorization of n.

All the terms are even by definition.

Links

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

Crossrefs

Cf. A000290, A001222, A162641, A268335, A350387, A350388.

Programs

Mathematica

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

PARI

a(n) = my(f=factor(n)); sum(k=1, #f~, if (!(f[k, 2] % 2), f[k, 2])); \\ Michel Marcus, Dec 29 2021

Python

from sympy import factorint
def a(n): return sum(e for e in factorint(n).values() if e%2 == 0)
print([a(n) for n in range(1, 106)]) # Michael S. Branicky, Dec 28 2021

Formula

Additive with a(p^e) = e if e is even and 0 otherwise.

a(n) = A001222(A350388(n)).

a(n) = 0 if and only if n is an exponentially odd number (A268335).

a(n) = A001222(n) - A350387(n).

a(n) = A001222(n) if and only if n is a positive square (A000290 \ {0}).

Asymptotic mean: lim_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 2*p/((p-1)*(p+1)^2) = 0.7961706018...