A362412 The number of primes factors of the square root of the largest square dividing n, counted with multiplicity.

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

Offset

1,16

Links

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

Formula

a(n) = A001222(A000188(n)).

a(n) = A001222(A008833(n))/2.

Additive with a(p^e) = floor(e/2) = A004526(e).

a(n) >= 0, with equality if and only if n is squarefree (A005117).

a(n) <= A001222(n)/2, with equality if and only if n is square (A000290).

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{p prime} 1/(p^2-1) = 0.551693... (A154945).

Mathematica

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

Prog

(PARI) a(n) = vecsum(factor(n)[, 2]\2);

Crossrefs

Cf. A000188, A000290, A001222, A004526, A005117, A008833, A154945, A349326.

Sequence in context: A158971 A121467 A366073 * A376657 A071325 A064727

Adjacent sequences: A362409 A362410 A362411 * A362413 A362414 A362415

Keyword

nonn,easy

Author

Amiram Eldar, Sep 28 2023

Status

approved