A183094 a(n) = number of powerful divisors d (excluding 1) of n.

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 3, 0, 1, 0, 1, 0, 0, 0, 2, 1, 0, 2, 1, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 3, 1, 1, 0, 1, 0, 2, 0, 2, 0, 0, 0, 1, 0, 0, 1, 5, 0, 0, 0, 1, 0, 0, 0, 5, 0, 0, 1, 1, 0, 0, 0, 3, 3, 0, 0, 1, 0, 0, 0, 2, 0, 1, 0, 1, 0, 0, 0, 4, 0, 1, 1, 3, 0, 0, 0, 2, 0

Offset

1,8

Comments

a(n) = number of divisors d of n from set A001694(m) - powerful numbers for m >=2.

Links

Robert Israel, Table of n, a(n) for n = 1..10000

D. Suryanarayana and R. Sitaramachandra Rao, The number of square-full divisors of an integer, Proc. Amer. Math. Soc. 34 (1972), 79-80.

Formula

a(n) = A000005(n) - A183095(n) = A005361(n) - 1.

a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = k-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = zeta(2)*zeta(3)/zeta(6) - 1 = A082695 - 1 = 0.9435964368... . - Amiram Eldar, Jul 30 2022

Example

For n = 12, set of such divisors is {4}; a(12) = 1.

Maple

f:=  n -> convert(map(t->t[2], ifactors(n)[2]), `*`) - 1; # Robert Israel, Jul 14 2014

Mathematica

powerfulQ[n_] := Min[ Last@# & /@ FactorInteger[n]] > 1; f[n_] := Length@ Select[ Divisors@ n, powerfulQ]; Array[f, 105] (* Robert G. Wilson v, Jul 14 2014 *)

Crossrefs

Cf. A000005, A001694, A005361, A082695, A183095.

Sequence in context: A066301 A046660 A368251 * A373968 A324192 A108730

Adjacent sequences: A183091 A183092 A183093 * A183095 A183096 A183097

Keyword

nonn

Author

Jaroslav Krizek, Dec 25 2010

Status

approved