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

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

Author

Amiram Eldar, Oct 18 2020

Keywords

• nonn
• cons

Comments

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...

Examples

			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...
		

References

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

Links

• 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.

Crossrefs

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

Programs

Mathematica

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

Formula

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)).