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%I #33 Apr 03 2024 03:10:42
%S 3,5,2,7,0,0,0,4,7,1,8,5,2,9,5,2,8,2,9,7,6,1,5,3,6,7,9,1,7,6,9,3,2,6,
%T 2,0,3,7,6,3,5,6,4,3,4,4,9,5,2,4,0,8,2,7,7,6,0,5,7,1,7,8,2,0,6,1,9,2,
%U 1,5,4,6,3,8,0,4,1,8,8,6,1,4,8,2,3,4,1
%N Decimal expansion of Sum_{n >= 1} sigma_1(n)/n!.
%C Problem No. 45 from P. Erdős (see the first reference). The problem is "is Sum_{n >= 1} sigma_k(n)/n! an irrational number where sigma_k(n) is the sum of the k-th power of divisors of n?" This property has been proved with k = 1 and 2 (see the second reference for the proof).
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1961-22.pdf">Some unsolved problems</a>, Publ. Inst. Hung. Acad. Sci. 6 (1961), 221-259.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1963-14.pdf">Quelques problèmes de théorie des nombres</a> (in French), Monographies de l'Enseignement Mathématique, No. 6, pp. 81-135, L'Enseignement Mathématique, Université, Geneva, 1963.
%H P. Erdős, <a href="http://www.renyi.hu/~p_erdos/1988-22.pdf">On the irrationality of certain series: problems and results</a>, in New advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp. 102-109.
%H P. Erdős & M. Kac, <a href="http://www.jstor.org/stable/2306485">Problem 4518</a>, Amer. Math. Monthly 60(1953) 47. <a href="http://www.jstor.org/stable/2306405">Solution</a> R. Breusch, 61 (1954) 264-265.
%e 3.52700047185295282976153...
%p with(numtheory):Digits:=200: s:=evalf(sum('sigma(i)/i!', 'i'=1..500)):print(s):
%t RealDigits[N[Sum[DivisorSigma[1,n]/n!, {n, 0, 500}], 200]][[1]]
%o (PARI) suminf(n=1, sigma(n)/n!) \\ _Michel Marcus_, Sep 16 2017
%Y Cf. A000203, A227989.
%K nonn,cons
%O 1,1
%A _Michel Lagneau_, Aug 02 2013
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