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A183097
a(n) = sum of powerful divisors d (including 1) of n.
17
1, 1, 1, 5, 1, 1, 1, 13, 10, 1, 1, 5, 1, 1, 1, 29, 1, 10, 1, 5, 1, 1, 1, 13, 26, 1, 37, 5, 1, 1, 1, 61, 1, 1, 1, 50, 1, 1, 1, 13, 1, 1, 1, 5, 10, 1, 1, 29, 50, 26, 1, 5, 1, 37, 1, 13, 1, 1, 1, 5, 1, 1, 10, 125, 1, 1, 1, 5, 1, 1, 1, 130, 1, 1, 26, 5, 1, 1, 1, 29, 118, 1, 1, 5, 1, 1, 1, 13, 1, 10, 1, 5, 1, 1, 1, 61, 1, 50, 10, 130
OFFSET
1,4
COMMENTS
Sequence is not the same as A091051(n); a(72) = 130, A091051(72) = 58.
a(n) = sum of divisors d of n from set A001694 - powerful numbers.
FORMULA
a(n) = A000203(n) - A183098(n) = A183100(n) + 1.
a(1) = 1, a(p) = 1, a(pq) = 1, a(pq...z) = 1, a(p^k) = ((p^(k+1)-1) / (p-1))-p, for p, q = primes, k = natural numbers, pq...z = product of k (k > 2) distinct primes p, q, ..., z.
From Amiram Eldar, Dec 24 2023: (Start)
Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)^2/(3*zeta(3)) = 1.892451... . (End)
EXAMPLE
For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.
MAPLE
A183097 := proc(n)
local a, pe, p, e ;
a := 1;
for pe in ifactors(n)[2] do
p := op(1, pe) ;
e := op(2, pe) ;
if e > 1 then
a := a* ( (p^(e+1)-1)/(p-1)-p) ;
end if;
end do:
a ;
end proc: # R. J. Mathar, Jun 02 2020
MATHEMATICA
fun[p_, e_] := (p^(e+1)-1)/(p-1) - p; a[1] = 1; a[n_] := Times @@ (fun @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, May 14 2019 *)
PROG
(PARI) A183097(n) = sumdiv(n, d, ispowerful(d)*d); \\ Antti Karttunen, Oct 07 2017
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, (f[i, 1]^(f[i, 2]+1)-1) / (f[i, 1]-1) - f[i, 1]); } \\ Amiram Eldar, Dec 24 2023
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Dec 25 2010
STATUS
approved