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 (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.
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
KEYWORD
nonn
AUTHOR
William J. Keith, Jul 17 2008, Jul 18 2008
STATUS
approved