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A183099
a(n) = sum of powerful divisors d (excluding 1) of n.
2
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, 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, 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
OFFSET
1,4
COMMENTS
a(n) = sum of divisors d of n from set A001694(m) - powerful numbers for m >=2.
FORMULA
a(n) = A000203(n) - A183100(n) = A183097(n) - 1.
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.
EXAMPLE
For n = 12, set of such divisors is {4}; a(12) = 4.
MATHEMATICA
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 *)
PROG
(PARI) A183099(n) = (sumdiv(n, d, ispowerful(d)*d) - 1); \\ Antti Karttunen, Oct 07 2017
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Jaroslav Krizek, Dec 25 2010
STATUS
approved