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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.

%I A183093 #21 Feb 12 2025 19:04:44
%S A183093 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,
%T A183093 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,
%U A183093 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
%N 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.
%C A183093 a(n) = number of non-powerful divisors d of n where powerful numbers are numbers of the form A001694(m) for m >= 2.
%C A183093 Sequence is not the same as A183096: a(72) = 6, A183096(72) = 7.
%H A183093 Antti Karttunen, <a href="/A183093/b183093.txt">Table of n, a(n) for n = 1..16384</a>
%H A183093 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%F A183093 a(n) = A000005(n) - A005361(n) = A183095(n) - 1.
%F A183093 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.
%F A183093 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
%e A183093 For n = 12, set of such divisors is {2, 3, 6, 12}; a(12) = 4.
%t A183093 nonpowQ[n_] := Min[FactorInteger[n][[;;, 2]]] == 1; a[n_] := DivisorSum[n, 1 &, # > 1 && nonpowQ[#] &]; Array[a, 100] (* _Amiram Eldar_, Jan 30 2025 *)
%o A183093 (Scheme) (define (A183093 n) (- (A000005 n) (A005361 n))) ;; _Antti Karttunen_, Nov 23 2017
%o A183093 (PARI) a(n) = sumdiv(n, d, d > 1 && vecmin(factor(d)[, 2]) == 1); \\ _Amiram Eldar_, Jan 30 2025
%Y A183093 Cf. A000005, A001694, A005361, A183094, A183095, A183096.
%Y A183093 Cf. A001620, A082695.
%K A183093 nonn
%O A183093 1,6
%A A183093 _Jaroslav Krizek_, Dec 25 2010
%E A183093 Name corrected by _Amiram Eldar_, Jan 30 2025