A183095 a(n) = number of divisors d of n that are either 1 or of the form Product_(i) (p_i^e_i) where e_i = 1 for at least one i.

1, 2, 2, 2, 2, 4, 2, 2, 2, 4, 2, 5, 2, 4, 4, 2, 2, 5, 2, 5, 4, 4, 2, 6, 2, 4, 2, 5, 2, 8, 2, 2, 4, 4, 4, 6, 2, 4, 4, 6, 2, 8, 2, 5, 5, 4, 2, 7, 2, 5, 4, 5, 2, 6, 4, 6, 4, 4, 2, 11, 2, 4, 5, 2, 4, 8, 2, 5, 4, 8, 2, 7, 2, 4, 5, 5, 4, 8, 2, 7, 2, 4, 2, 11, 4, 4, 4, 6, 2, 11, 4, 5, 4, 4, 4, 8, 2, 5, 5, 6

Offset: 1

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn

Comments

a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 1.

Examples

			For n = 12, set of such divisors is {1, 2, 3, 6, 12}; a(12) = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..10000
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A001694, A183093, A183094.

Cf. A001620, A082695.

Programs

Mathematica

nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1;a[n_] := DivisorSum[n, 1 &, nonpowQ[#] &]; Array[a, 100] (* Amiram Eldar, Jan 30 2025 *)

PARI

a(n) = sumdiv(n, d, d == 1 || vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025

Formula

a(n) = A000005(n) - A183094(n) = A183093(n) + 1.

a(1) = 1, a(p) = 2, a(pq) = 4, a(pq...z) = 2^k, a(p^k) = 2, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.

Sum_{k=1..n} a(k) ~ n*(log(n) + 2*gamma - zeta(2)*zeta(3)/zeta(6)), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Extensions

Name corrected by Amiram Eldar, Jan 30 2025