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

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

Author

Jaroslav Krizek, Dec 25 2010

Keywords

• nonn
• mult

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.

Examples

			For n = 12, set of such divisors is {1, 4}; a(12) = 1+4 = 5.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16385
• Index entries for sequences related to sums of divisors.

Crossrefs

Cf. A001694, A091051, A183102.

Cf. A002117, A078434.

Programs

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 *)

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

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)