A143029 A sequence of asymptotic density zeta(3) - 1, where zeta is the Riemann zeta function.

3, 11, 14, 19, 27, 32, 35, 43, 51, 59, 67, 68, 75, 76, 78, 83, 86, 91, 99, 107, 115, 122, 123, 131, 139, 140, 147, 155, 163, 171, 172, 174, 176, 179, 187, 194, 195, 203, 211, 219, 227, 230, 235, 243, 248, 251, 259, 267, 268, 270, 275, 283, 284, 291, 299, 302

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 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 congruent to m-1 mod m and floor(x/(m*(m!))) is not congruent to m-1 mod m. The sequence is made up of the residue classes 3 mod 8; 14 and 32 mod 54; 76, 78, 172, 174, 268 and 270 mod 384, etc. A set of such sequences with entries for each zeta(k) - 1 partitions the integers. See the linked paper for their construction.

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. A143028, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.

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[300], f[#] == 3 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)

Formula

a(n) = 2*a(n-1) + 3. [Obviously wrong, R. J. Mathar, Jul 14 2016]

G.f.: 1/(exp(x)-1). [Apparently not, R. J. Mathar, Jul 14 2016]