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a(n) = n^2 * Sum_{d^2|n} 1 / d^2.
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%I #33 Mar 24 2023 08:23:25

%S 1,4,9,20,25,36,49,80,90,100,121,180,169,196,225,336,289,360,361,500,

%T 441,484,529,720,650,676,810,980,841,900,961,1344,1089,1156,1225,1800,

%U 1369,1444,1521,2000,1681,1764,1849,2420,2250,2116,2209,3024,2450,2600,2601,3380,2809

%N a(n) = n^2 * Sum_{d^2|n} 1 / d^2.

%H Seiichi Manyama, <a href="/A351600/b351600.txt">Table of n, a(n) for n = 1..10000</a>

%F G.f.: Sum_{k>=1} k^2 * x^(k^2) * (1 + x^(k^2)) / (1 - x^(k^2))^3. - _Ilya Gutkovskiy_, Feb 21 2022

%F Multiplicative with a(p^e) = p^2*(p^(2*e) - p^(2*floor((e-1)/2)))/(p^2 - 1). - _Sebastian Karlsson_, Feb 25 2022

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = zeta(4)/3 = Pi^4/270 = 0.360774... . - _Amiram Eldar_, Nov 13 2022

%t f[p_, e_] := p^2*(p^(2*e) - p^(2*Floor[(e - 1)/2]))/(p^2 - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* _Amiram Eldar_, Nov 13 2022 *)

%o (PARI) a(n) = n^2*sumdiv(n, d, if (issquare(d), 1/d)); \\ _Michel Marcus_, Feb 15 2022

%Y Sequences of the form n^k * Sum_{d^2|n} 1/d^k for k = 0..10: A046951 (k=0), A340774 (k=1), this sequence (k=2), A351601 (k=3), A351602 (k=4), A351603 (k=5), A351604 (k=6), A351605 (k=7), A351606 (k=8), A351607 (k=9), A351608 (k=10).

%Y Cf. A013662, A076752.

%K nonn,mult

%O 1,2

%A _Wesley Ivan Hurt_, Feb 14 2022