

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.


20



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
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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
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 nth 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), 187213.


LINKS

Peter Luschny, Table of n, a(n) for n = 1..20000 (first 10000 terms from T. D. Noe)
J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, arXiv:math/0008177 [math.NT], 20002001; Am. Math. Monthly 109 (#6, 2002), 534543.
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

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


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

Cf. A057640, A000203, A076633, A067698, A079526, A058209.
Sequence in context: A342360 A251967 A330725 * A336029 A272876 A133930
Adjacent sequences: A057638 A057639 A057640 * A057642 A057643 A057644


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



