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

1,6

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.

Links

Antti Karttunen, Table of n, a(n) for n = 1..16384

Index entries for sequences computed from exponents in factorization of n.

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

Example

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

Mathematica

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

Prog

(Scheme) (define (A183093 n) (- (A000005 n) (A005361 n))) ;; Antti Karttunen, Nov 23 2017
(PARI) a(n) = sumdiv(n, d, d > 1 && vecmin(factor(d)[, 2]) == 1); \\ Amiram Eldar, Jan 30 2025

Crossrefs

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

Cf. A001620, A082695.

Sequence in context: A372720 A095960 A316314 * A183096 A377071 A029356

Adjacent sequences: A183090 A183091 A183092 * A183094 A183095 A183096

Keyword

nonn,changed

Author

Jaroslav Krizek, Dec 25 2010

Extensions

Name corrected by Amiram Eldar, Jan 30 2025

Status

approved