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

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

Author

Amiram Eldar, Dec 15 2022

Keywords

• nonn
• frac

Examples

			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, ...
		

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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).