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

1,4

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

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...

Mathematica

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

Prog

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

Crossrefs

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

Sequence in context: A171919 A000089 A339430 * A352550 A348326 A364213

Adjacent sequences: A350383 A350384 A350385 * A350387 A350388 A350389

Keyword

nonn,easy

Author

Amiram Eldar, Dec 28 2021

Status

approved