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A335341 Sum of divisors of A003557(n).

%I A335341 #27 Nov 02 2023 04:48:32
%S A335341 1,1,1,3,1,1,1,7,4,1,1,3,1,1,1,15,1,4,1,3,1,1,1,7,6,1,13,3,1,1,1,31,1,
%T A335341 1,1,12,1,1,1,7,1,1,1,3,4,1,1,15,8,6,1,3,1,13,1,7,1,1,1,3,1,1,4,63,1,
%U A335341 1,1,3,1,1,1,28,1,1,6,3,1,1,1
%N A335341 Sum of divisors of A003557(n).
%C A335341 The sum of the divisors d of n such that n/d is a coreful divisor of n (a coreful divisor of n is a divisor with the same squarefree kernel as n). The number of these divisors is A005361(n). - _Amiram Eldar_, Jun 30 2023
%H A335341 Antti Karttunen, <a href="/A335341/b335341.txt">Table of n, a(n) for n = 1..16383</a>
%H A335341 Antti Karttunen, <a href="/A335341/a335341.txt">Data supplement: n, a(n) computed for n = 1..65537</a>
%F A335341 a(n) = A000203(A003557(n)).
%F A335341 Multiplicative with a(p^1)=1 and a(p^e) = (p^e-1)/(p-1) if e>1.
%F A335341 A057723(n) = A007947(n)*a(n).
%F A335341 a(n) = 1 iff n in A005117.
%F A335341 a(n) = A336567(n) + A003557(n). - _Antti Karttunen_, Jul 28 2020
%F A335341 Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-1)). - _Amiram Eldar_, Sep 09 2023
%F A335341 a(n) = A047994(n)/A173557(n). - _Ridouane Oudra_, Oct 30 2023
%p A335341 A335341 := proc(n)
%p A335341     local a,pe,p,e ;
%p A335341     a := 1;
%p A335341     for pe in ifactors(n)[2] do
%p A335341         p := op(1,pe) ;
%p A335341         e := op(2,pe) ;
%p A335341         if e > 1 then
%p A335341             a := a*(p^e-1)/(p-1) ;
%p A335341         end if;
%p A335341     end do:
%p A335341     a ;
%p A335341 end proc:
%t A335341 f[p_, e_] := (p^e-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 26 2020 *)
%o A335341 (PARI) a(n) = sigma(n/factorback(factor(n)[, 1])); \\ _Michel Marcus_, Jun 02 2020
%Y A335341 Cf. A000203, A003557, A005361 (number of divisors of A003557), A336567.
%Y A335341 Cf. A047994, A173557.
%K A335341 nonn,mult,easy
%O A335341 1,4
%A A335341 _R. J. Mathar_, Jun 02 2020