This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A351266 #34 Feb 09 2025 01:11:29
%S A351266 1,9,28,9,126,252,344,9,28,1134,1332,252,2198,3096,3528,9,4914,252,
%T A351266 6860,1134,9632,11988,12168,252,126,19782,28,3096,24390,31752,29792,9,
%U A351266 37296,44226,43344,252,50654,61740,61544,1134,68922,86688,79508,11988,3528,109512,103824
%N A351266 Sum of the cubes of the squarefree divisors of n.
%C A351266 Inverse Möbius transform of n^3 * mu(n)^2. - _Wesley Ivan Hurt_, Jun 08 2023
%H A351266 Seiichi Manyama, <a href="/A351266/b351266.txt">Table of n, a(n) for n = 1..10000</a>
%H A351266 N. J. A. Sloane, <a href="/transforms.txt">Transforms</a>.
%F A351266 a(n) = Sum_{d|n} d^3 * mu(d)^2.
%F A351266 a(n) = abs(A328640(n)).
%F A351266 G.f.: Sum_{k>=1} mu(k)^2 * k^3 * x^k / (1 - x^k). - _Ilya Gutkovskiy_, Feb 06 2022
%F A351266 Multiplicative with a(p^e) = 1 + p^3. - _Amiram Eldar_, Feb 06 2022
%F A351266 Sum_{k=1..n} a(k) ~ c * n^4, where c = zeta(4)/(4*zeta(2)) = Pi^2/60 = 0.164493... . - _Amiram Eldar_, Nov 10 2022
%F A351266 Dirichlet g.f.: zeta(s)*zeta(s-3)/zeta(2s-6). - _Michael Shamos_, Feb 09 2025
%e A351266 a(4) = 9; a(4) = Sum_{d|4} d^3 * mu(d)^2 = 1^3*1 + 2^3*1 + 4^3*0 = 9.
%t A351266 a[1] = 1; a[n_] := Times @@ (1 + FactorInteger[n][[;; , 1]]^3); Array[a, 100] (* _Amiram Eldar_, Feb 06 2022 *)
%o A351266 (PARI) a(n) = sumdiv(n, d, if (issquarefree(d), d^3)); \\ _Michel Marcus_, Feb 06 2022
%Y A351266 Cf. A008683 (mu), A013661, A013662, A328640.
%Y A351266 Sum of the k-th powers of the squarefree divisors of n for k=0..10: A034444 (k=0), A048250 (k=1), A351265 (k=2), this sequence (k=3), A351267 (k=4), A351268 (k=5), A351269 (k=6), A351270 (k=7), A351271 (k=8), A351272 (k=9), A351273 (k=10).
%K A351266 nonn,mult
%O A351266 1,2
%A A351266 _Wesley Ivan Hurt_, Feb 05 2022