

A143028


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


10



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, 37, 38, 40, 41, 42, 44, 45, 46, 48, 49, 52, 53, 54, 56, 57, 58, 60, 61, 62, 64, 65, 66, 69, 70, 72, 73, 74, 77, 80, 81, 82, 84, 85, 88, 89, 90, 92, 93, 94, 96, 97, 98, 100, 101, 102, 105
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OFFSET

1,2


COMMENTS

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 (m2)!, floor[x/(m!)] is not congruent to m1 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.
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


LINKS

Amiram Eldar, Table of n, a(n) for n = 1..10000
William J. Keith, Sequences of Density zeta(K)  1, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.


MATHEMATICA

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 *)


CROSSREFS

Cf. A143029, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A339013.
Cf. A161189.  Gary W. Adamson, Jun 07 2009
Sequence in context: A006594 A172276 A260483 * A277018 A277008 A091529
Adjacent sequences: A143025 A143026 A143027 * A143029 A143030 A143031


KEYWORD

nonn


AUTHOR

William J. Keith, Jul 17 2008, Jul 18 2008


STATUS

approved



