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.

A336065 Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).

This page as a plain text file.
%I A336065 #9 Jul 08 2020 05:08:49
%S A336065 8,4,8,9,5,7,1,9,5,0,0,4,4,9,3,3,2,8,1,4,2,7,1,0,9,7,6,8,5,4,4,3,5,2,
%T A336065 9,2,6,7,7,9,1,4,7,2,8,9,9,4,9,1,8,1,0,0,9,7,8,8,1,7,6,4,4,2,0,5,6,1,
%U A336065 5,7,0,9,6,6,9,2,4,6,7,0,3,0,0,1,5,8,6
%N A336065 Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).
%D A336065 József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.
%H A336065 Andrzej Schinzel and Tibor Šalát, <a href="https://dml.cz/handle/10338.dmlcz/136624">Remarks on maximum and minimum exponents in factoring</a>, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.
%F A336065 Equals 1/zeta(2) + Sum_{k>=2} ((1/zeta(k+1)) * Product_{p prime, p|k} ((p^(k-e(p,k)+1) - 1)/(p^(k+1) - 1)) - (1/zeta(k)) * Product_{p prime, p|k} ((p^(k-e(p,k)) - 1)/(p^k - 1))), where e(p,k) is the largest exponent of p dividing k.
%e A336065 0.848957195004493328142710976854435292677914728994918...
%t A336065 f[k_] := Module[{f = FactorInteger[k]}, p = f[[;; , 1]]; e = f[[;; , 2]]; (1/Zeta[k + 1])* Times @@ ((p^(k - e + 1) - 1)/(p^(k + 1) - 1)) - (1/Zeta[k]) * Times @@ ((p^(k - e) - 1)/(p^k - 1))]; RealDigits[1/Zeta[2] + Sum[f[k], {k, 2, 1000}], 10, 100][[1]]
%Y A336065 Cf. A051903, A059956, A336064.
%K A336065 nonn,cons
%O A336065 0,1
%A A336065 _Amiram Eldar_, Jul 07 2020