A369427 The number of unitary divisors of n that are squares of primes.

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

Offset

1,36

Comments

The number of exponents in the prime factorization of n that are equal to 2.

Links

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

Formula

Additive with a(p^e) = 1 if e = 2, and 0 otherwise.

a(n) > 0 if and only if n is in A038109.

a(A061742(n)) = n, and a(k) < n for all k < A061742(n).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} (1/p^2 - 1/p^3) = A085548 - A085541 = 0.27748478074162196208... .

Mathematica

f[p_, e_] := If[e == 2, 1, 0]; a[1] = 0; a[n_] := Plus @@ f @@@ FactorInteger[n]; Array[a, 100]

Prog

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

Crossrefs

Cf. A001248, A038109, A056169, A056170, A061742, A085541, A085548.

Similar sequences: A125070, A293439, A366074, A367512, A369426, A369428.

Sequence in context: A193138 A255320 A256574 * A304819 A304820 A162641

Adjacent sequences: A369424 A369425 A369426 * A369428 A369429 A369430

Keyword

nonn,easy

Author

Amiram Eldar, Jan 23 2024

Status

approved