cp's OEIS Frontend

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.

A190623 Mobius transform of A008457.

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%I A190623 #15 Dec 03 2022 05:45:57
%S A190623 1,6,27,64,125,162,343,512,729,750,1331,1728,2197,2058,3375,4096,4913,
%T A190623 4374,6859,8000,9261,7986,12167,13824,15625,13182,19683,21952,24389,
%U A190623 20250,29791,32768,35937,29478,42875,46656,50653,41154,59319,64000,68921,55566,79507,85184,91125,73002,103823,110592,117649
%N A190623 Mobius transform of A008457.
%C A190623 Multiplicative because A008457 is. - _Andrew Howroyd_, Jul 23 2018
%D A190623 J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 195.
%H A190623 Amiram Eldar, <a href="/A190623/b190623.txt">Table of n, a(n) for n = 1..10000</a>
%F A190623 From _Amiram Eldar_, Dec 03 2022: (Start)
%F A190623 Multiplicative with a(2) = 6, a(2^e) = 8^e for e > 1, and a(p^e) = p^(3*e) for p > 2.
%F A190623 Dirichlet g.f.: zeta(s-3)*(1 - 2^(1-s) + 4^(2-s)).
%F A190623 Sum_{k=1..n} a(k) ~ (15/64) * n^4. (End)
%t A190623 b[n_] := (-1)^n Sum[(-1)^d d^3, {d, Divisors[n]}];
%t A190623 a[n_] := Sum[MoebiusMu[d] b[n/d], {d, Divisors[n]}];
%t A190623 Array[a, 49] (* _Jean-François Alcover_, Sep 07 2019, from PARI *)
%t A190623 f[p_, e_] := p^(3*e); f[2, 1] = 6; f[2, e_] := 8^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Dec 03 2022 *)
%o A190623 (PARI) \\ here b(n) is A008457.
%o A190623 b(n)=(-1)^n*sumdiv(n, d, (-1)^d*d^3);
%o A190623 a(n)=sumdiv(n, d, moebius(d)*b(n/d)); \\ _Andrew Howroyd_, Jul 23 2018
%Y A190623 Cf. A008457.
%K A190623 nonn,mult
%O A190623 1,2
%A A190623 _N. J. A. Sloane_, May 14 2011