%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