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 A365498 #26 Jan 27 2025 06:50:48
%S A365498 1,2,2,2,2,4,2,1,2,4,2,4,2,4,4,1,2,4,2,4,4,4,2,2,2,4,1,4,2,8,2,1,4,4,
%T A365498 4,4,2,4,4,2,2,8,2,4,4,4,2,2,2,4,4,4,2,2,4,2,4,4,2,8,2,4,4,1,4,8,2,4,
%U A365498 4,8,2,2,2,4,4,4,4,8,2,2,1,4,2,8,4,4,4
%N A365498 Dirichlet g.f.: zeta(s) * Product_{p prime} (1 + 1/p^s - 1/p^(3*s)).
%C A365498 The number of unitary divisors of n that are cubefree numbers (A004709). - _Amiram Eldar_, Sep 06 2023
%H A365498 Vaclav Kotesovec, <a href="/A365498/b365498.txt">Table of n, a(n) for n = 1..10000</a>
%H A365498 Jon Maiga, <a href="http://sequencedb.net/s/A365498">Computer-generated formulas for A365498</a>, Sequence Machine.
%F A365498 Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
%F A365498 Let f(s) = Product_{p prime} (1 - 1/p^(2*s) - 1/p^(3*s) + 1/p^(4*s)).
%F A365498 Sum_{k=1..n} a(k) ~ f(1) * n * (log(n) + 2*gamma - 1 + f'(1)/f(1)), where
%F A365498 f(1) = Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.5358961538283379998085026313185459506482223745141452711510108346133288...,
%F A365498 f'(1) = f(1) * Sum_{p prime} (-4 + 3*p + 2*p^2) * log(p) / (1 - p - p^2 + p^4) = f(1) * 1.4525924794451595590371439593828547341482465114411929136723476679...
%F A365498 and gamma is the Euler-Mascheroni constant A001620.
%F A365498 Multiplicative with a(p^e) = 2 if e <= 2, and 1 otherwise. - _Amiram Eldar_, Sep 06 2023
%F A365498 From _Vaclav Kotesovec_, Jan 27 2025: (Start)
%F A365498 Following formulas have been conjectured for this sequence by Sequence Machine, with each one giving the first 1000000 terms correctly:
%F A365498 a(n) = A056671(n) * A368885(n).
%F A365498 a(n) = A034444(n) / A368248(n).
%F A365498 a(n) = A158522(n) / A307428(n).
%F A365498 a(n) = A369310(n) / A190867(n).
%F A365498 a(n) = A286324(n) / A368172(n). (End)
%t A365498 f[p_, e_] := If[e <= 2, 2, 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Sep 06 2023 *)
%o A365498 (PARI) for(n=1, 100, print1(direuler(p=2, n, 1/(1-X) * (1 + X - X^3))[n], ", "))
%Y A365498 Cf. A001620, A004709, A056671, A365488, A365499.
%Y A365498 Cf. A034444, A158522, A190867, A286324, A307428, A368172, A368248, A368885, A369310.
%K A365498 nonn,easy,mult
%O A365498 1,2
%A A365498 _Vaclav Kotesovec_, Sep 06 2023