A183093 a(n) = number of divisors d of n such that d > 1 and if d = Product_(i) (p_i^e_i) then e_i = 1 for at least one i.

0, 1, 1, 1, 1, 3, 1, 1, 1, 3, 1, 4, 1, 3, 3, 1, 1, 4, 1, 4, 3, 3, 1, 5, 1, 3, 1, 4, 1, 7, 1, 1, 3, 3, 3, 5, 1, 3, 3, 5, 1, 7, 1, 4, 4, 3, 1, 6, 1, 4, 3, 4, 1, 5, 3, 5, 3, 3, 1, 10, 1, 3, 4, 1, 3, 7, 1, 4, 3, 7, 1, 6, 1, 3, 4, 4, 3, 7, 1, 6, 1, 3, 1, 10, 3, 3, 3, 5, 1, 10, 3, 4, 3, 3, 3, 7, 1, 4, 4, 5

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 >= 2.

Sequence is not the same as A183096: a(72) = 6, A183096(72) = 7.

Examples

			For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences computed from exponents in factorization of n.

Crossrefs

Cf. A000005, A001694, A005361, A183094, A183095, A183096.

Cf. A001620, A082695.

Programs

Mathematica

nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1; a[n_] := DivisorSum[n, 1 &, # > 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

Scheme

(define (A183093 n) (- (A000005 n) (A005361 n))) ;; Antti Karttunen, Nov 23 2017

Formula

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

a(1) = 0, a(p) = 1, a(pq) = 3, a(pq...z) = 2^k - 1, a(p^k) = 1, 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) - 1), where gamma is Euler's constant (A001620). - Amiram Eldar, Jan 30 2025

Extensions

Name corrected by Amiram Eldar, Jan 30 2025