A380266 a(n) is the numerator of the mean value of A051904(k) at the range k = 1..n.

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 5, 16, 17, 6, 11, 23, 4, 25, 13, 9, 14, 29, 5, 32, 33, 4, 37, 38, 13, 40, 45, 46, 47, 48, 25, 51, 26, 53, 27, 55, 4, 57, 29, 59, 30, 61, 31, 64, 13, 22, 67, 68, 23, 14, 71, 24, 73, 74, 5, 76, 77, 26, 21, 17, 43, 87, 22, 89, 9

Offset: 1

Author

Amiram Eldar, Jan 18 2025

Keywords

• nonn
• easy
• frac

Examples

			Fractions begin with 0, 1/2, 2/3, 1, 1, 1, 1, 5/4, 4/3, 13/10, 14/11, 5/4, ...
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
• Eric Weisstein's World of Mathematics, Niven's Constant.
• Wikipedia, Niven's constant.

Crossrefs

Cf. A051904, A090699, A380267 (denominators).

Programs

Mathematica

f[n_] := Min[FactorInteger[n][[;;, 2]]]; f[1] = 0; With[{m = 100}, Numerator[Accumulate[Array[f, m]] / Range[m]]]

PARI

lista(nmax) = {my(s = 0); print1(0, ", "); for(n = 2, nmax, s += vecmin(factor(n)[,2]);  print1(numerator(s/n), ", "));}

Formula

a(n) = numerator((Sum_{k=1..n} A051904(k))/n).

a(n)/A380267(n) = 1 + c/sqrt(n) + o(1/sqrt(n)), where c = zeta(3/2)/zeta(3) (A090699).