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A351265 Sum of the squares of the squarefree divisors of n.

%I A351265 #37 Dec 26 2024 12:00:32
%S A351265 1,5,10,5,26,50,50,5,10,130,122,50,170,250,260,5,290,50,362,130,500,
%T A351265 610,530,50,26,850,10,250,842,1300,962,5,1220,1450,1300,50,1370,1810,
%U A351265 1700,130,1682,2500,1850,610,260,2650,2210,50,50,130,2900,850,2810,50,3172,250,3620
%N A351265 Sum of the squares of the squarefree divisors of n.
%C A351265 Inverse Möbius transform of n^2 * mu(n)^2. - _Wesley Ivan Hurt_, Jun 08 2023
%H A351265 Seiichi Manyama, <a href="/A351265/b351265.txt">Table of n, a(n) for n = 1..10000</a>
%H A351265 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A351265 a(n) = Sum_{d|n} d^2 * mu(d)^2.
%F A351265 a(n) = abs(A328639(n)).
%F A351265 G.f.: Sum_{k>=1} mu(k)^2 * k^2 * x^k / (1 - x^k). - _Ilya Gutkovskiy_, Feb 06 2022
%F A351265 Multiplicative with a(p^e) = 1 + p^2. - _Amiram Eldar_, Feb 06 2022
%F A351265 Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(3)/(3*zeta(2)) = A253905 / 3 = 0.243587... . - _Amiram Eldar_, Nov 10 2022
%F A351265 Dirichlet g.f.: zeta(s)*zeta(s-2)/zeta(2s-4). - _Michael Shamos_, Aug 05 2023
%e A351265 a(6) = 50; a(6) = Sum_{d|6} d^2 * mu(d)^2 = 1^2*1 + 2^2*1 + 3^2*1 + 6^2*1 = 50.
%t A351265 a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^2); Array[a, 100] (* _Amiram Eldar_, Feb 06 2022 *)
%t A351265 Table[Total[Select[Divisors[n],SquareFreeQ]^2],{n,80}] (* _Harvey P. Dale_, Dec 26 2024 *)
%o A351265 (PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^2)); \\ _Michel Marcus_, Feb 06 2022
%Y A351265 Cf. A008683 (mu), A253905, A328639, A322360.
%Y A351265 Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), this sequence (k=2), A351266 (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
%K A351265 nonn,mult
%O A351265 1,2
%A A351265 _Wesley Ivan Hurt_, Feb 05 2022