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

1,1

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.

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]

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 *)

Crossrefs

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

Sequence in context: A101585 A115214 A373955 * A186701 A022123 A303035

Adjacent sequences: A143026 A143027 A143028 * A143030 A143031 A143032

Keyword

nonn

Author

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

Status

approved