A350387 a(n) is the sum of the odd exponents in the prime factorization of n.

0, 1, 1, 0, 1, 2, 1, 3, 0, 2, 1, 1, 1, 2, 2, 0, 1, 1, 1, 1, 2, 2, 1, 4, 0, 2, 3, 1, 1, 3, 1, 5, 2, 2, 2, 0, 1, 2, 2, 4, 1, 3, 1, 1, 1, 2, 1, 1, 0, 1, 2, 1, 1, 4, 2, 4, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 3, 1, 2, 1, 1, 2, 3, 1, 1, 0, 2, 1, 2, 2, 2, 2, 4, 1, 2, 2, 1, 2, 2, 2, 6, 1, 1, 1, 0, 1, 3, 1, 4, 3

Offset

1,6

Comments

First differs from A125073 at n = 32.

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

Links

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

Formula

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

a(n) = A001222(A350389(n)).

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

a(n) = A001222(n) - A350386(n).

a(n) = A001222(n) if and only if n is an exponentially odd number (A268335).

Sum_{k=1..n} a(k) = n * log(log(n)) + c * n + O(n/log(n)), where c = A083342 - Sum_{p prime} 2*p/((p-1)*(p+1)^2) = gamma + Sum_{p prime} (log(1-1/p) + (p^2+1)/((p-1)*(p+1)^2)) = 0.2384832800... and gamma is Euler's constant (A001620).

Mathematica

f[p_, e_] := If[OddQ[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 == 1)
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 28 2021

Crossrefs

Cf. A000290, A001222, A001620, A083342, A125073, A162641, A268335, A350386, A350389.

Sequence in context: A321377 A071467 A125073 * A325310 A308881 A281488

Adjacent sequences: A350384 A350385 A350386 * A350388 A350389 A350390

Keyword

nonn,easy

Author

Amiram Eldar, Dec 28 2021

Status

approved