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

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Comments

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

Examples

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

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.

Crossrefs

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

Programs

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 *)

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