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Numbers k such that exp(H_k)*log(H_k) <= sigma(k), where H_k is the harmonic number.
0

%I #29 Feb 14 2019 10:06:45

%S 1,2,3,4,6,12,24,60

%N Numbers k such that exp(H_k)*log(H_k) <= sigma(k), where H_k is the harmonic number.

%C If the Riemann hypothesis is true, there are no more terms.

%H J. C. Lagarias, <a href="https://arxiv.org/abs/math/0008177">An elementary problem equivalent to the Riemann hypothesis</a>, arXiv:math/0008177 [math.NT], 2000-2001; Am. Math. Monthly 109 (#6, 2002), 534-543.

%e Let b(n) = exp(H_{a(n)})*log(H_{a(n)}).

%e n | a(n) | b(n) | sigma(a(n))

%e --+------+------------+-------------

%e 1 | 1 | 0 | 1

%e 2 | 2 | 1.817... | 3

%e 3 | 3 | 3.791... | 4

%e 4 | 4 | 5.894... | 7

%e 5 | 6 | 10.384... | 12

%e 6 | 12 | 25.218... | 28

%e 7 | 24 | 57.981... | 60

%e 8 | 60 | 166.296... | 168

%t For[k = 1, True, k++, If[Exp[HarmonicNumber[k]] Log[HarmonicNumber[k]] <= DivisorSigma[1, k], Print[k]]] (* _Jean-François Alcover_, Feb 14 2019 *)

%Y Cf. A000203, A067698, A079526, A079527.

%K nonn,more

%O 1,2

%A _Seiichi Manyama_, Feb 09 2019