cp's OEIS Frontend

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

%I A143029 #19 Feb 15 2021 04:30:52
%S A143029 3,11,14,19,27,32,35,43,51,59,67,68,75,76,78,83,86,91,99,107,115,122,
%T A143029 123,131,139,140,147,155,163,171,172,174,176,179,187,194,195,203,211,
%U A143029 219,227,230,235,243,248,251,259,267,268,270,275,283,284,291,299,302
%N A143029 A sequence of asymptotic density zeta(3) - 1, where zeta is the Riemann zeta function.
%C A143029 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.
%H A143029 Amiram Eldar, <a href="/A143029/b143029.txt">Table of n, a(n) for n = 1..10000</a>
%H A143029 William J. Keith, <a href="https://www.emis.de/journals/INTEGERS/papers/k19/k19.Abstract.html">Sequences of Density zeta(K) - 1</a>, INTEGERS, Vol. 10 (2010), Article #A19, pp. 233-241. Also <a href="http://arxiv.org/abs/0905.3765">arXiv preprint</a>, arXiv:0905.3765 [math.NT], 2009 and <a href="http://www.math.drexel.edu/~keith/ZetaKMinusOne.pdf">author's copy</a>.
%F A143029 a(n) = 2*a(n-1) + 3. [Obviously wrong, _R. J. Mathar_, Jul 14 2016]
%F A143029 G.f.: 1/(exp(x)-1). [Apparently not, _R. J. Mathar_, Jul 14 2016]
%t A143029 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 *)
%Y A143029 Cf. A143028, A143030, A143031, A143032, A143033, A143034, A143035, A143036, A161189, A339013.
%K A143029 nonn
%O A143029 1,1
%A A143029 _William J. Keith_, Jul 17 2008, Jul 18 2008