%I
%S 7,23,39,50,55,71,87,103,104,119,135,151,167,183,199,212,215,231,247,
%T 263,266,279,295,311,327,343,359,364,366,374,375,391,407,423,428,439,
%U 455,471,487,503,519,535,536,551,567,583,590,599,615,631,647,663,679
%N A sequence of asymptotic density zeta(4)  1, where zeta is the Riemann zeta function.
%C x is an element of this sequence if when m is the least natural number such that the least positive residue of x mod m! is no more than (m2)!, floor(x/(m!)) and floor(x/(m*(m!))) are congruent to m1 mod m, but floor(x/((m^2)*(m!))) is not. The sequence is made up of the residue classes 7 (mod 16); 50 and 104 (mod 162); 364, 366, 748, 750, 1132 and 1134 (mod 1536), etc. A set of such sequences with entries for each zeta(k)  1 partitions the integers. See the linked paper for their construction.
%H Amiram Eldar, <a href="/A143030/b143030.txt">Table of n, a(n) for n = 1..10000</a>
%H William J. Keith, <a href="https://www.emis.de/journals/INTEGERS/papers/k19/k19.Abstract.html">Sequences of Density zeta(K)  1</a>, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233241. Also <a href="http://arxiv.org/abs/0905.3765">arXiv preprint</a>, arXiv:0905.3765 [math.NT], 2009 and <a href="http://www.math.drexel.edu/~keith/ZetaKMinusOne.pdf">author's copy</a>.
%t f[n_] := Module[{k = n  1, m = 2, r}, While[{k, r} = QuotientRemainder[k, m]; r != 0, m++]; IntegerExponent[k + 1, m] + 2]; Select[Range[700], f[#] == 4 &] (* _Amiram Eldar_, Feb 15 2021 after _Kevin Ryde_ at A161189 *)
%Y Cf. A143028, A143029, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.
%K nonn
%O 1,1
%A _William J. Keith_, Jul 18 2008
