A207709 - OEIS

%I #25 Feb 16 2025 08:33:16

%S 1,1,1,1,1,1,1,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,

%T 1,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,2,1,2,1,2,1,1,1,2,1,2,1,

%U 2,1,2,1,2,1,1,1,2,1,2,1,1,1,2,1,2,1,2

%N Floor((H(n) + exp(H(n))*log(H(n)))/sigma(n)), where H(n) is the harmonic number sum_{i=1..n} 1/i.

%C An assertion equivalent to the Riemann hypothesis is: a(n) > 0 for every n >= 1.

%C a(12*n) = 1 for all 1<=n<=43312.

%C For n >= 1, a(2^(10^n)) so far appears to equal floor(n*(exp(1)-2/3) - 1/3).

%C There exist integers n such that (H(n) + exp(H(n))*log(H(n)))/sigma(n) < 1.01 (i.e., n = 100630609505753353981293837481689271234222794240000*1087#). See A215640 for information on how to generate these numbers. - _Arkadiusz Wesolowski_, Aug 19 2012

%H Arkadiusz Wesolowski, <a href="/A207709/b207709.txt">Table of n, a(n) for n = 1..10000</a>

%H J. C. Lagarias, <a href="http://arXiv.org/abs/math.NT/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543.

%H Eric W. Weisstein, <a href="https://mathworld.wolfram.com/RiemannHypothesis.html">MathWorld: Riemann Hypothesis</a>

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Jeffrey_Lagarias">Jeffrey Lagarias</a>

%e a(11) = 2 because (H(11) + exp(1)^H(11)*log(H(11)))/sigma(11) = 2.1387006307....

%t lst = {}; Do[h = NIntegrate[(1 - x^n)/(1 - x), {x, 0, 1}]; AppendTo[lst, Floor[(h + Exp@h*Log@h)/DivisorSigma[1, n]]], {n, 530}]; lst

%Y Cf. A057641, A000203, A008594, A076633.

%K nonn,changed

%O 1,11

%A _Arkadiusz Wesolowski_, Feb 19 2012