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%I #15 Aug 29 2023 04:20:31
%S 0,0,0,4,0,0,0,12,9,0,0,4,0,0,0,28,0,9,0,4,0,0,0,12,25,0,36,4,0,0,0,
%T 60,0,0,0,49,0,0,0,12,0,0,0,4,9,0,0,28,49,25,0,4,0,36,0,12,0,0,0,4,0,
%U 0,9,124,0,0,0,4,0,0,0,129,0,0,25,4,0,0,0,28,117,0,0,4,0,0,0,12,0,9,0,4,0,0,0,60,0,49,9,129
%N a(n) = sum of powerful divisors d (excluding 1) of n.
%C a(n) = sum of divisors d of n from set A001694(m) - powerful numbers for m >=2.
%H Antti Karttunen, <a href="/A183099/b183099.txt">Table of n, a(n) for n = 1..16385</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F a(n) = A000203(n) - A183100(n) = A183097(n) - 1.
%F a(1) = 0, a(p) = 0, a(pq) = 0, a(pq...z) = 0, a(p^k) = ((p^(k+1)-1) / (p-1))-p-1, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
%e For n = 12, set of such divisors is {4}; a(12) = 4.
%t f[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 0; a[n_] := Times @@ f @@@ FactorInteger[n] - 1; Array[a, 100] (* _Amiram Eldar_, Aug 29 2023 *)
%o (PARI) A183099(n) = (sumdiv(n, d, ispowerful(d)*d) - 1); \\ _Antti Karttunen_, Oct 07 2017
%Y Cf. A000203, A001694, A183097, A183100.
%K nonn,easy
%O 1,4
%A _Jaroslav Krizek_, Dec 25 2010
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