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

Author

Amiram Eldar, Jul 07 2020

Keywords

• nonn
• cons

Examples

			0.848957195004493328142710976854435292677914728994918...
		

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.

Crossrefs

Cf. A051903, A059956, A336064.

Programs

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]]

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.