A227989 Decimal expansion of Sum_{n >= 1} sigma_2(n)/n!.

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

Offset: 1

Author

Michel Lagneau, Aug 02 2013

Keywords

• nonn
• cons

Comments

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).

Examples

			6.3400966688921716388299...
		

References

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

Links

• P. Erdős, Some unsolved problems, Publ. Inst. Hung. Acad. Sci. 6 (1961), 221-259.
• P. Erdős, Quelques problèmes de théorie des nombres (in French), Monographies de l'Enseignement Mathématique, No. 6, pp. 81-135, L'Enseignement Mathématique, Université de Genève, 1963.
• P. Erdős, On the irrationality of certain series: problems and results, in New advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp.102-109.
• P. Erdős & M. Kac, Problem 4518, Amer. Math. Monthly 60(1953) 47. Solution R. Breusch, 61 (1954) 264-265.

Crossrefs

Cf. A001157, A227988.

Programs

Mathematica

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