This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A335532 #23 Feb 16 2025 08:34:00
%S A335532 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,
%T A335532 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,
%U A335532 8,1,5,9,3,0,3,2,5,0,3,8,7,7,0,0,3,4,1
%N A335532 Decimal expansion of the asymptotic value of the second raw moment of the maximal exponent in the prime factorizations of n (A051903).
%C A335532 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 <H> = 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 <H^2> = Sum_{k>=1} k^2*d(k), whose simplified formula in terms of zeta functions is given in the FORMULA section.
%C A335532 The second central moment, or variance, of H is <H^2> - <H>^2 = 4.3013024003... - 1.7052111401...^2 = 1.3935573679... and the standard deviation is sqrt(<H^2> - <H>^2) = 1.1804903082...
%D A335532 Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.6 Niven's constant, pp. 112-113.
%H A335532 Ivan Niven, <a href="https://doi.org/10.1090/S0002-9939-1969-0241373-5">Averages of Exponents in Factoring Integers</a>, Proc. Amer. Math. Soc., Vol. 22, No. 2 (1969), pp. 356-360.
%H A335532 D. Suryanarayana and R. Sita Rama Chandra Rao, <a href="https://doi.org/10.1007/BF01223919">On the maximum and minimum exponents in factoring integers</a>, Archiv der Mathematik, Vol. 28, No. 1 (1977), pp. 261-269.
%H A335532 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/NivensConstant.html">Niven's Constant</a>.
%H A335532 Wikipedia, <a href="https://en.wikipedia.org/wiki/Niven%27s_constant">Niven's constant</a>.
%F A335532 Equals lim_{n->oo} (1/n) * Sum_{k=1..n} A051903(k)^2.
%F A335532 Equals 1 + Sum_{j>=2} (2*j-1) * (1 - 1/zeta(j)).
%e A335532 4.30130240031336659998068934041877579922989129763477...
%e A335532 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...
%t A335532 RealDigits[1 + Sum[(2*j - 1)*(1 - 1/Zeta[j]), {j, 2, 400}], 10, 100][[1]]
%Y A335532 Cf. A005117, A033150, A051903, A067259, A242977.
%K A335532 nonn,cons
%O A335532 1,1
%A A335532 _Amiram Eldar_, Oct 18 2020