A279609 - OEIS

A279609

a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).

%I #13 Dec 25 2016 15:36:46

%S 0,0,0,2,6,34,207,492,9051,143828,306310,963859,5155084,81053635,

%T 1334916490,29106956400,58655156200,1817551636000,56466287472000,

%U 376943525488000,1144451930851200,41803526752345600

%N a(n) = floor(H(k) + exp(H(k))*log(H(k))) - sigma(k) where H(k) is the k-th harmonic number Sum_{j=1..k} 1/j and k is the n-th colossally abundant number A004490(n).

%C By a theorem of J. C. Lagarias, the Riemann hypothesis is equivalent to the proposition that this sequence never takes a negative value. In fact, by inspection it appears to be monotone increasing; this conjecture implies the Riemann hypothesis but is not in any obvious way implied by it. Stronger conjectures are easy to formulate--for example, if F(n) is the function defined by this sequence, then F(n)/2^n also appears to be monotone increasing.

%H J. C. Lagarias, <a href="http://www.jstor.org/stable/2695443">An elementary problem equivalent to the Riemann hypothesis</a>, Am. Math. Monthly 109 (#6, 2002), 534-543.

%Y Cf. A004490, A057641.

%K nonn

%O 2,4

%A _Gene Ward Smith_, Dec 15 2016