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%I #29 Oct 13 2023 06:52:42
%S 1,1,1,3,1,1,1,3,4,1,1,3,1,1,1,3,1,4,1,3,1,1,1,3,6,1,4,3,1,1,1,3,1,1,
%T 1,12,1,1,1,3,1,1,1,3,4,1,1,3,8,6,1,3,1,4,1,3,1,1,1,3,1,1,4,3,1,1,1,3,
%U 1,1,1,12,1,1,6,3,1,1,1,3,4,1,1,3,1,1,1,3,1,4,1,3,1,1,1,3,1,8,4,18,1,1,1,3,1
%N Sum of squarefree divisors of the powerful part of n: a(n) = A048250(A057521(n)).
%C The sum of the squarefree divisors of n whose square divides n. - _Amiram Eldar_, Oct 13 2023
%H Antti Karttunen, <a href="/A295295/b295295.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+1) for e > 1.
%F a(n) = A048250(A057521(n)) = A048250(A064549(A003557(n))).
%F a(n) = A048250(n) / A092261(n).
%F a(n) = Sum_{d^2|n} d * mu(d)^2. - _Wesley Ivan Hurt_, Feb 13 2022
%F From _Amiram Eldar_, Sep 18 2023: (Start)
%F Dirichlet g.f.: zeta(s) * zeta(2*s-1) / zeta(4*s-2).
%F Sum_{k=1..n} a(k) ~ (3*n/Pi^2) * (log(n) + 3*gamma - 1 - 4*zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). (End)
%F a(n) = A048250(n) - A344137(n). - _Amiram Eldar_, Oct 13 2023
%t Array[DivisorSum[#/Denominator[#/Apply[Times, FactorInteger[#][[All, 1]]]^2], # &, SquareFreeQ] &, 105] (* _Michael De Vlieger_, Nov 26 2017, after _Jean-François Alcover_ at A057521 *)
%t f[p_, e_] := If[e == 1, 1, p+1] ; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 18 2023 *)
%o (Scheme, with memoization-macro definec)
%o (definec (A295295 n) (if (= 1 n) n (let ((p (A020639 n)) (e (A067029 n))) (* (if (= 1 e) 1 (+ 1 p)) (A295295 (A028234 n))))))
%o (PARI) a(n) = my(f=factor(n)); for (i=1, #f~, if (f[i,2]==1, f[i,1]=1)); sumdiv(factorback(f), d, d*issquarefree(d)); \\ _Michel Marcus_, Jan 29 2021
%Y Cf. A000203, A003557, A048250, A057521, A064549, A092261, A295294, A344137.
%Y Cf. A001620, A013661, A073002.
%K nonn,easy,mult
%O 1,4
%A _Antti Karttunen_, Nov 25 2017
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