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%I #10 Apr 29 2019 20:51:05
%S 1,4,6,9,3,5,3,2,8,4,7,2,6,9,2,8,2,2,3,3,1,2,3,5,5,1,3,6,4,9,8,2,0,5,
%T 6,6,3,8,8,6,3,1,9,3,9,5,7,6,7,0,3,4,5,8,4,8,5,8,3,8,8,1,7,0,5,3,1,5,
%U 3,0,6,2,6,3,5,4,5,9,9,5,3,7,4,4,0,1,8
%N Decimal expansion of Sum_{k >= 1} sigma_3(k)/k!, where sigma_3(k) is the sum of cubes of the divisors of k (A001158).
%C Its irrationality was conjectured by Erdős and Kac in 1953 and was proved by Schlage-Puchta in 2006 and Friedlander et al. in 2007.
%H Paul Erdős and Mark Kac, <a href="http://www.jstor.org/stable/2306485">Problem 4518</a>, American Mathematical Monthly, Vol. 60, No. 1 (1953), p. 47.
%H John B. Friedlander, Florian Luca, and Mihai Stoiciu, <a href="http://www.emis.de/journals/INTEGERS/papers/h31/h31.Abstract.html">On the irrationality of a divisor function series</a>, Integers, Vol. 7, No. 1 (2007), A31.
%H Jan-Christoph Schlage-Puchta, <a href="https://doi.org/10.1007/s11139-006-0154-3">The irrationality of a number theoretical series</a>, The Ramanujan Journal, Vol. 12, No. 3 (2006), pp. 455-460, <a href="https://arxiv.org/abs/1105.1452">preprint</a>, arXiv:1105.1452 [math.NT], 2011.
%e 14.6935328472692822331235513649820566388631939576...
%t RealDigits[N[Sum[DivisorSigma[3, n]/n!, {n, 1, 500}], 100]][[1]]
%o (PARI) suminf(k=1, sigma(k, 3)/k!) \\ _Michel Marcus_, Mar 21 2019
%Y Cf. A001158, A227988, A227989.
%K nonn,cons
%O 2,2
%A _Amiram Eldar_, Mar 21 2019
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