OFFSET
1,4
COMMENTS
LINKS
FORMULA
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