A057641 a(n) = floor(H(n) + exp(H(n))*log(H(n))) - sigma(n), where H(n) = Sum_{k=1..n} 1/k and sigma(n) (A000203) is the sum of the divisors of n.

0, 0, 1, 0, 4, 0, 7, 2, 7, 5, 13, 0, 17, 9, 12, 8, 23, 5, 27, 8, 21, 20, 34, 1, 33, 25, 30, 17, 46, 7, 50, 22, 40, 37, 46, 6, 62, 43, 50, 19, 70, 19, 74, 37, 46, 55, 82, 9, 79, 46, 70, 47, 95, 32, 83, 38, 81, 74, 107, 2, 112, 81, 76, 56, 102, 45, 125, 70, 103, 58, 133, 14, 138, 101

Offset: 1

Author

N. J. A. Sloane, Oct 12 2000

Keywords

• nonn
• nice
• easy

Comments

Theorem (Lagarias): a(n) is nonnegative for all n if and only if the Riemann Hypothesis is true.

Up to rank n=10^4, zeros occur only at n=1,2,4,6 and 12; ones occur at n=3 and n=24. The first occurrence of k = 0,1,2,3,... is at n = 1,3,8,-1,5,10,36,7,16,14,-1,-1,15,11,72,... where -1 means that k does not occur among the first 10^4 terms. - Robert G. Wilson v, Dec 06 2010, reformulated by M. F. Hasler, Sep 09 2011

Looking at the graph of this sequence, it appears that there is a slowly growing lower bound. It is even more apparent when larger ranges of points are computed. Numbers A176679(n+2) and A222761(n) give the (x,y) coordinates of the n-th point. - T. D. Noe, Mar 28 2013

References

• G. Robin, Grandes valeurs de la fonction somme des diviseurs et hypothèse de Riemann, J. Math. Pures Appl. 63 (1984), 187-213.

Links

• Peter Luschny, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
• Masazumi Honda and Takuya Yoda, String theory, N = 4 SYM and Riemann hypothesis, arXiv:2203.17091 [hep-th], 2022.
• J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.
• S. Nazardonyavi and S. Yakubovich, Delicacy of the Riemann hypothesis and certain subsequences of superabundant numbers, arXiv preprint arXiv:1306.3434 [math.NT], 2013.

Crossrefs

Cf. A057640, A000203, A076633, A067698, A079526, A058209.

Programs

Mathematica

f[n_] := Block[{h = HarmonicNumber@n}, Floor[h + Exp@h*Log@h] - DivisorSigma[1, n]]; Array[f, 74] (* Robert G. Wilson v, Dec 06 2010 *)

PARI

a(n)={my(H=sum(k=1,n,1/k)); floor(exp(H)*log(H)+H) - sigma(n)}
list_A057641(Nmax,H=0,S=1)=for(n=S,Nmax, H+=1/n; print1(floor(exp(H)*log(H)+H) - sigma(n),","))  \\ M. F. Hasler, Sep 09 2011

Formula

a(n) = A057640(n) - A000203(n). - Omar E. Pol, Oct 25 2019

Extensions

Five more terms from Robert G. Wilson v, Dec 06 2010

I deleted some unproved assertions by Robert G. Wilson v about the presence of 0's, 1's, ... in this sequence. - N. J. A. Sloane, Dec 07 2010