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

1,6

Comments

First differs from A125070 at n = 64.

Links

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

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.

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]

Prog

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

Crossrefs

Cf. A000290, A010052, A077761, A197680.

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

Sequence in context: A056169 A286852 A341595 * A125070 A125071 A335699

Adjacent sequences: A369425 A369426 A369427 * A369429 A369430 A369431

Keyword

nonn,easy

Author

Amiram Eldar, Jan 23 2024

Status

approved