A143030 A sequence of asymptotic density zeta(4) - 1, where zeta is the Riemann zeta function.

7, 23, 39, 50, 55, 71, 87, 103, 104, 119, 135, 151, 167, 183, 199, 212, 215, 231, 247, 263, 266, 279, 295, 311, 327, 343, 359, 364, 366, 374, 375, 391, 407, 423, 428, 439, 455, 471, 487, 503, 519, 535, 536, 551, 567, 583, 590, 599, 615, 631, 647, 663, 679

Offset: 1

Author

William J. Keith, 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!)) and floor(x/(m*(m!))) are congruent to m-1 mod m, but floor(x/((m^2)*(m!))) is not. The sequence is made up of the residue classes 7 (mod 16); 50 and 104 (mod 162); 364, 366, 748, 750, 1132 and 1134 (mod 1536), 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, A143029, 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[700], f[#] == 4 &] (* Amiram Eldar, Feb 15 2021 after Kevin Ryde at A161189 *)