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

Author

Amiram Eldar, Dec 28 2021

Keywords

• nonn
• easy

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

Crossrefs

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

Programs

Mathematica

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

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

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).