

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.


22



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



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



EXTENSIONS



STATUS

approved



