OFFSET
1,5
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
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.
FORMULA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,nice,easy
AUTHOR
N. J. A. Sloane, Oct 12 2000
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
STATUS
approved