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

1,1

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!)) 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.

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 *)

Crossrefs

Cf. A143028, A143029, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.

Sequence in context: A098039 A132237 A227064 * A031043 A183126 A213632

Adjacent sequences: A143027 A143028 A143029 * A143031 A143032 A143033

Keyword

nonn

Author

William J. Keith, Jul 18 2008

Status

approved