A380264 a(n) is the numerator of the mean value of A051903(k) at the range k = 1..n.

0, 1, 2, 1, 1, 1, 1, 5, 4, 13, 14, 4, 17, 9, 19, 23, 24, 13, 27, 29, 10, 31, 32, 35, 37, 19, 41, 43, 44, 3, 46, 51, 52, 53, 54, 14, 57, 29, 59, 31, 63, 32, 65, 67, 23, 35, 71, 25, 11, 79, 80, 41, 83, 43, 87, 45, 91, 46, 93, 19, 96, 97, 11, 105, 106, 107, 108, 55

Offset

1,3

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.

Formula

a(n) = numerator((Sum_{k=1..n} A051903(k))/n).

a(n)/A380265(n) = A129132(n)/n.

Limit_{n->oo} a(n)/A380265(n) = c, where c is Niven's constant (A033150).

abs(a(n)/A380265(n) - c) <= 3*log_2(n)/sqrt(n).

Example

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

Mathematica

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

Prog

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

Crossrefs

Cf. A033150, A051903, A129132, A359071, A359072, A380265 (denominators).

Sequence in context: A327858 A307372 A259681 * A380266 A146314 A202480

Adjacent sequences: A380261 A380262 A380263 * A380265 A380266 A380267

Keyword

nonn,easy,frac

Author

Amiram Eldar, Jan 18 2025

Status

approved