A242977 -id:A242977 - cp's OEIS frontend

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

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).

A335532 Decimal expansion of the asymptotic value of the second raw moment of the maximal exponent in the prime factorizations of n (A051903).

Original entry on oeis.org

4, 3, 0, 1, 3, 0, 2, 4, 0, 0, 3, 1, 3, 3, 6, 6, 5, 9, 9, 9, 8, 0, 6, 8, 9, 3, 4, 0, 4, 1, 8, 7, 7, 5, 7, 9, 9, 2, 2, 9, 8, 9, 1, 2, 9, 7, 6, 3, 4, 7, 7, 4, 3, 1, 6, 4, 7, 3, 8, 6, 9, 9, 1, 7, 2, 7, 2, 4, 8, 1, 5, 9, 3, 0, 3, 2, 5, 0, 3, 8, 7, 7, 0, 0, 3, 4, 1

Offset: 1

Amiram Eldar, Oct 18 2020

Let H(n) = A051903(n) be the maximal exponent in the prime factorizations of n. The asymptotic density of the numbers whose maximal exponent is k is d(k) = 1/zeta(k+1) - 1/z(k). For example, k=1 corresponds to the squarefree numbers (A005117), and k=2 corresponds to the cubefree numbers which are not squarefree (A067259). The asymptotic mean of H is = Sum_{k>=1} k*d(k) = 1 + Sum_{j>=2} (1 - 1/zeta(j)) = 1.705211... which is Niven's constant (A033150). The second raw moment of the distribution of maximal exponents is = Sum_{k>=1} k^2*d(k), whose simplified formula in terms of zeta functions is given in the FORMULA section.

The second central moment, or variance, of H is - ^2 = 4.3013024003... - 1.7052111401...^2 = 1.3935573679... and the standard deviation is sqrt( - ^2) = 1.1804903082...

			4.30130240031336659998068934041877579922989129763477...
For the numbers n=1..2^20, the values of H(n) = A051903(n) are in the range [0..20]. Their mean value is 894015/524288 = 1.705198..., their second raw moment is 140939/32768 = 4.301116..., and their standard deviation is sqrt(383019202687/274877906944) = 1.180430...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, pp. 112-113.

Ivan Niven, Averages of Exponents in Factoring Integers, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
D. Suryanarayana and R. Sita Rama Chandra Rao, On the maximum and minimum exponents in factoring integers, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
Eric Weisstein's World of Mathematics, Niven's Constant.
Wikipedia, Niven's constant.

Cf. A005117, A033150, A051903, A067259, A242977.

RealDigits[1 + Sum[(2*j - 1)*(1 - 1/Zeta[j]), {j, 2, 400}], 10, 100][[1]]

Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A051903(k)^2.

Equals 1 + Sum_{j>=2} (2*j-1) * (1 - 1/zeta(j)).

cp's OEIS Frontend

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

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