A359072 -id:A359072 - cp's OEIS frontend

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

Amiram Eldar, Jan 18 2025

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

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.

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

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

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

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).

A359071 Numerators of the partial sums of the reciprocals of the maximal exponent in prime factorization of the positive integers (A051903).

Original entry on oeis.org

1, 2, 5, 7, 9, 11, 35, 19, 22, 25, 53, 59, 65, 71, 145, 157, 163, 175, 181, 193, 205, 217, 221, 227, 239, 81, 83, 87, 91, 95, 479, 499, 519, 539, 549, 569, 589, 609, 1847, 1907, 1967, 2027, 2057, 2087, 2147, 2207, 1111, 563, 1141, 1171, 593, 608, 613, 628, 211

Offset: 2

Amiram Eldar, Dec 15 2022

			Fractions begin with 1, 2, 5/2, 7/2, 9/2, 11/2, 35/6, 19/3, 22/3, 25/3, 53/6, 59/6, ...

Amiram Eldar, Table of n, a(n) for n = 2..10000
Wolfgang Schwarz and Jürgen Spilker, A remark on some special arithmetical functions, in: E. Laurincikas , E. Manstavicius and V. Stakenas (eds.), Analytic and Probabilistic Methods in Number Theory, Proceedings of the Second International Conference in Honour of J. Kubilius, Palanga, Lithuania, 23-27 September 1996, New Trends in Probability and Statistics, Vol. 4, VSP BV & TEV Ltd. (1997), pp. 221-245.
D. Suryanarayana and R. Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Index entries for sequences computed from exponents in factorization of n.

Cf. A051903, A129132, A242977, A359072 (denominators).

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

a(n) = numerator(Sum_{k=2..n} 1/A051903(k)).

a(n)/A359072(n) = c_1 * n + O(n^(1/2)*exp(-c_2*log(n)^(3/5)/log(log(n))^(1/5))), where c_1 = A242977 and c_2 is a constant, 0 < c_2 < 1/2^(8/5) (Suryanarayana and R. Chandra Rao, 1977).

cp's OEIS Frontend

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

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