cp's OEIS Frontend

A183097 a(n) = sum of powerful divisors d (including 1) of n.

%I A183097 #23 Dec 24 2023 01:07:30
%S A183097 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,
%T A183097 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,
%U A183097 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
%N A183097 a(n) = sum of powerful divisors d (including 1) of n.
%C A183097 Sequence is not the same as A091051(n); a(72) = 130,  A091051(72) = 58.
%C A183097 a(n) = sum of divisors d of n from set A001694 - powerful numbers.
%H A183097 Antti Karttunen, <a href="/A183097/b183097.txt">Table of n, a(n) for n = 1..16385</a>
%H A183097 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F A183097 a(n) = A000203(n) - A183098(n) = A183100(n) + 1.
%F A183097 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.
%F A183097 From _Amiram Eldar_, Dec 24 2023: (Start)
%F A183097 Dirichlet g.f.: zeta(s) * zeta(2*s-2) * zeta(3*s-3) / zeta(6*s-6).
%F A183097 Sum_{k=1..n} a(k) ~ c * n^(3/2), where c = zeta(3/2)^2/(3*zeta(3)) = 1.892451... . (End)
%e A183097 For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.
%p A183097 A183097 := proc(n)
%p A183097     local a,pe,p,e ;
%p A183097     a := 1;
%p A183097     for pe in ifactors(n)[2] do
%p A183097         p := op(1,pe) ;
%p A183097         e := op(2,pe) ;
%p A183097         if e > 1 then
%p A183097             a := a* ( (p^(e+1)-1)/(p-1)-p) ;
%p A183097         end if;
%p A183097     end do:
%p A183097     a ;
%p A183097 end proc: # _R. J. Mathar_, Jun 02 2020
%t A183097 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 *)
%o A183097 (PARI) A183097(n) = sumdiv(n, d, ispowerful(d)*d); \\ _Antti Karttunen_, Oct 07 2017
%o A183097 (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
%Y A183097 Cf. A001694, A091051, A183102.
%Y A183097 Cf. A002117, A078434.
%K A183097 nonn,mult
%O A183097 1,4
%A A183097 _Jaroslav Krizek_, Dec 25 2010