A369428 The number of exponents in the prime factorization of n that are squares.

0, 1, 1, 0, 1, 2, 1, 0, 0, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 2, 2, 1, 1, 0, 2, 0, 1, 1, 3, 1, 0, 2, 2, 2, 0, 1, 2, 2, 1, 1, 3, 1, 1, 1, 2, 1, 2, 0, 1, 2, 1, 1, 1, 2, 1, 2, 2, 1, 2, 1, 2, 1, 0, 2, 3, 1, 1, 2, 3, 1, 0, 1, 2, 1, 1, 2, 3, 1, 2, 1, 2, 1, 2, 2, 2, 2

Offset: 1

Author

Amiram Eldar, Jan 23 2024

Keywords

• nonn
• easy

Comments

First differs from A125070 at n = 64.

Links

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

Crossrefs

Cf. A000290, A010052, A077761, A197680.

Similar sequences: A125070, A293439, A366074, A367512, A369426, A369427.

Programs

Maple

a:= n-> add(`if`(issqr(i[2]), 1, 0), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 23 2024

Mathematica

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

PARI

a(n) = vecsum(apply(x -> if(issquare(x), 1, 0), factor(n)[, 2]));

Formula

Additive with a(p^e) = A010052(e).

Sum_{k=1..n} a(k) ~ n * (log(log(n)) + B - C), where B is Mertens's constant (A077761) and C = P(2) + Sum_{k>=2} (P(k^2+1) - P(k^2)) = 0.40999077396414387641..., and P(s) is the prime zeta function.