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

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

Table of n, a(n) for n=0..86.

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

Cf. A051903, A059956, A336064.

Sequence in context: A087015 A200224 A345929 * A124012 A335354 A000803

Adjacent sequences: A336062 A336063 A336064 * A336066 A336067 A336068

Keyword

nonn,cons

Author

Amiram Eldar, Jul 07 2020

Status

approved