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A176679 Conjecturally, numbers j for which f(m) > f(j) for all m > j, where f(k) = H(k) + exp(H(k))*log(H(k)) - sigma(k).

%I A176679 #17 Oct 13 2024 17:41:41
%S A176679 1,2,12,24,60,120,180,240,360,420,840,2520,5040,7560,10080,15120,
%T A176679 20160,27720,30240,32760,55440,65520,83160,110880,166320,196560,
%U A176679 221760,277200,332640,360360,393120,415800,720720,831600,942480,1441440,2162160
%N A176679 Conjecturally, numbers j for which f(m) > f(j) for all m > j, where f(k) = H(k) + exp(H(k))*log(H(k)) - sigma(k).
%C A176679 Here H(k) is the k-th harmonic number, Sum_{i=1..k} 1/i, and sigma(k) is the sum of the divisors of k. This function is derived from an inequality in Lagarias's paper. The condition f(k) > 0 for all k > 1 is equivalent to the Riemann hypothesis (RH). Every colossally abundant number (A004490) is here. Every superabundant number (A004394) greater than 665280 appears to be here also. This sequence is meaningless if the RH is false.
%H A176679 T. D. Noe, <a href="/A176679/b176679.txt">Table of n, a(n) for n = 1..84</a>
%H A176679 Jeffrey C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Amer. Math. Monthly 109 (#6, 2002), 534-543.
%t A176679 (* This is just a naive recomputation of a dozen terms. *)
%t A176679 H = HarmonicNumber;
%t A176679 f[k_] := H[k] + Exp[H[k]] Log[H[k]] - DivisorSigma[1, k];
%t A176679 okQ[k_] := AllTrue[Range[k+1, 2k], f[#] > f[k]&];
%t A176679 Reap[For[k = 1, k < 10^4, k = If[k >= 60, k+60, k+1], If[okQ[k], Print[k]; Sow[k]]]][[2, 1]] (* _Jean-François Alcover_, Dec 11 2018 *)
%Y A176679 Cf. A057641, A222761.
%K A176679 nonn
%O A176679 1,2
%A A176679 _T. D. Noe_, Apr 23 2010