A227989 - OEIS

%I #46 Sep 29 2024 06:27:59

%S 6,3,4,0,0,9,6,6,6,8,8,9,2,1,7,1,6,3,8,8,2,9,9,6,5,9,9,4,0,0,7,5,0,4,

%T 6,0,7,8,6,3,6,4,4,3,3,5,5,9,8,9,0,1,7,8,5,4,3,9,9,6,0,6,1,1,5,9,3,7,

%U 0,8,7,7,1,3,8,6,4,2,6,5,6,1,5,8,3,3,9

%N Decimal expansion of Sum_{n >= 1} sigma_2(n)/n!.

%C Problem No. 45 from P. Erdős (see the 1963 link). The problem is "is Sum_{n = 1..oo} 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 Breusch link for the proof).

%D R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section B14.

%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 de Genève, 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 6.3400966688921716388299...

%t RealDigits[N[Sum[DivisorSigma[2,n]/n!, {n, 0, 500}], 105]][[1]]

%Y Cf. A001157, A227988.

%K nonn,cons

%O 1,1

%A _Michel Lagneau_, Aug 02 2013