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A336065
Decimal expansion of the asymptotic density of the numbers divisible by the maximal exponent in their prime factorization (A336064).
3
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, 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, 5, 7, 0, 9, 6, 6, 9, 2, 4, 6, 7, 0, 3, 0, 0, 1, 5, 8, 6
OFFSET
0,1
REFERENCES
József Sándor and Borislav Crstici, Handbook of Number theory II, Kluwer Academic Publishers, 2004, chapter 3, p. 331.
LINKS
Andrzej Schinzel and Tibor Šalát, Remarks on maximum and minimum exponents in factoring, Mathematica Slovaca, Vol. 44, No. 5 (1994), pp. 505-514.
FORMULA
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.
EXAMPLE
0.848957195004493328142710976854435292677914728994918...
MATHEMATICA
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]]
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Amiram Eldar, Jul 07 2020
STATUS
approved