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).

1, 2, 12, 24, 60, 120, 180, 240, 360, 420, 840, 2520, 5040, 7560, 10080, 15120, 20160, 27720, 30240, 32760, 55440, 65520, 83160, 110880, 166320, 196560, 221760, 277200, 332640, 360360, 393120, 415800, 720720, 831600, 942480, 1441440, 2162160

Offset: 1

Author

T. D. Noe, Apr 23 2010

Keywords

• nonn

Comments

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.

Links

• T. D. Noe, Table of n, a(n) for n = 1..84
• Jeffrey C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Amer. Math. Monthly 109 (#6, 2002), 534-543.

Crossrefs

Cf. A057641, A222761.

Programs

Mathematica

(* This is just a naive recomputation of a dozen terms. *)
H = HarmonicNumber;
f[k_] := H[k] + Exp[H[k]] Log[H[k]] - DivisorSigma[1, k];
okQ[k_] := AllTrue[Range[k+1, 2k], f[#] > f[k]&];
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 *)