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%I #27 Sep 09 2023 06:50:41
%S 1,1,1,7,1,1,1,15,13,1,1,7,1,1,1,31,1,13,1,7,1,1,1,15,31,1,40,7,1,1,1,
%T 63,1,1,1,91,1,1,1,15,1,1,1,7,13,1,1,31,57,31,1,7,1,40,1,15,1,1,1,7,1,
%U 1,13,127,1,1,1,7,1,1,1,195,1,1,31,7,1,1,1,31,121,1,1,7,1,1,1,15,1,13,1,7,1,1,1,63
%N Sum of the divisors of the powerful part of n: a(n) = A000203(A057521(n)).
%H Antti Karttunen, <a href="/A295294/b295294.txt">Table of n, a(n) for n = 1..16384</a>
%H <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>.
%F Multiplicative with a(p) = 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e > 1.
%F a(n) = A000203(A057521(n)) = A000203(A064549(A003557(n))).
%F a(n) = A000203(n) / A092261(n).
%F From _Amiram Eldar_, Oct 08 2022: (Start)
%F a(n) = 1 iff n is squarefree (A005117).
%F a(n) = A000203(n) iff n is powerful (A001694). (End)
%F Dirichlet g.f.: zeta(s) * zeta(s-1) * Product_{p prime} (1 - 1/p^(s-1) + 1/p^(2*s-2) + 1/p^(2*s-1) - 1/p^(3*s-2)). - _Amiram Eldar_, Sep 09 2023
%t Array[DivisorSigma[1, #/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2] ] &, 96] (* _Michael De Vlieger_, Nov 26 2017, after _Jean-François Alcover_ at A057521 *)
%t f[p_, e_] := If[e == 1, 1, (p^(e+1)-1)/(p-1)]; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 08 2022 *)
%o (Scheme)
%o (define (A295294 n) (A000203 (A057521 n)))
%o ;; With memoization-macro definec:
%o (definec (A295294 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= e 1) 1 (/ (- (expt p (+ 1 e)) 1) (- p 1))) (A295294 (A028234 n))))))
%o (PARI) a(n) = {my(f = factor(n), p = f[,1], e = f[,2]); prod(i=1, #p, if(e[i] == 1, 1, (p[i]^(e[i]+1)-1)/(p[i]-1)))}; \\ _Amiram Eldar_, Oct 08 2022
%o (Python)
%o from math import prod
%o from sympy import factorint
%o def A295294(n): return prod((p**(e+1)-1)//(p-1) for p, e in factorint(n).items() if e > 1) # _Chai Wah Wu_, Nov 14 2022
%Y Cf. A000203, A001694, A003557, A005117, A057521, A064549, A092261, A295295.
%K nonn,easy,mult
%O 1,4
%A _Antti Karttunen_, Nov 25 2017
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