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A057641 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. 16
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 (list; graph; refs; listen; history; text; internal format)
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 n-th point. - T. D. Noe, Mar 28 2013

REFERENCES

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

LINKS

T. D. Noe, Table of n, a(n) for n=1..10000

J. C. Lagarias, An elementary problem equivalent to the Riemann hypothesis, 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, 2013

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: A199071 A157698 A251967 * A272876 A133930 A077892

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

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Last modified March 28 02:00 EDT 2017. Contains 284182 sequences.