A143028 A sequence of asymptotic density zeta(2) - 1, where zeta is the Riemann zeta function.

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

Offset: 1

Author

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

Keywords

• nonn

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 (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.

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. 233-241. Also arXiv preprint, arXiv:0905.3765 [math.NT], 2009 and author's copy.

Crossrefs

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

Cf. A161189. - Gary W. Adamson, Jun 07 2009

Programs

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 *)