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A sequence of asymptotic density zeta(2) - 1, where zeta is the Riemann zeta function.
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%I #19 Feb 08 2022 21:59:28

%S 1,2,4,5,6,8,9,10,12,13,16,17,18,20,21,22,24,25,26,28,29,30,33,34,36,

%T 37,38,40,41,42,44,45,46,48,49,52,53,54,56,57,58,60,61,62,64,65,66,69,

%U 70,72,73,74,77,80,81,82,84,85,88,89,90,92,93,94,96,97,98,100,101,102,105

%N A sequence of asymptotic density zeta(2) - 1, where zeta is the Riemann zeta function.

%C x is an element of this sequence if when m>1 is the least natural number such that the least positive residue of x mod m! is no more than (m-2)!, floor[x/(m!)] is not congruent to m-1 mod m. The sequence is made up of the residue classes 1 mod 4; 2 and 8 mod 18; 4, 6, 28, 30, 52 and 54 mod 96, etc. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

%C A161189(n) = 2 if n is a term of this sequence. Similarly A161189(n) = 3, 4, 5, ... if n is in A143029, A143030, ...; such that the number system is partitioned into relative densities tending to (zeta(2) - 1), (zeta(3) - 1), ... such that Sum_{k>=2} (zeta(k) - 1) = 1.0. This implies that the density of 2's in A161189 tends to (zeta(2) - 1) = (Pi^2/6 - 1) = 0.644934... . - _Gary W. Adamson_, Jun 07 2009

%H Amiram Eldar, <a href="/A143028/b143028.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. 233-241. 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[100], f[#] == 2 &] (* _Amiram Eldar_, Feb 15 2021 after _Kevin Ryde_ at A161189 *)

%Y Cf. A143029, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A339013.

%Y Cf. A161189. - _Gary W. Adamson_, Jun 07 2009

%K nonn

%O 1,2

%A _William J. Keith_, Jul 17 2008, Jul 18 2008