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

1,2

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.

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 *)

Crossrefs

Cf. A057641, A222761.

Sequence in context: A077562 A307003 A146567 * A278407 A224923 A247086

Adjacent sequences: A176676 A176677 A176678 * A176680 A176681 A176682

Keyword

nonn

Author

T. D. Noe, Apr 23 2010

Status

approved