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%I #9 Jul 08 2020 05:08:49
%S 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 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 5,7,0,9,6,6,9,2,4,6,7,0,3,0,0,1,5,8,6
%N Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).
%D József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.
%H 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 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 0.848957195004493328142710976854435292677914728994918...
%t 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 Cf. A051903, A059956, A336064.
%K nonn,cons
%O 0,1
%A _Amiram Eldar_, Jul 07 2020
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