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A227989
Decimal expansion of Sum_{n >= 1} sigma_2(n)/n!.
5
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
1,1
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).
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.
EXAMPLE
6.3400966688921716388299...
MATHEMATICA
RealDigits[N[Sum[DivisorSigma[2, n]/n!, {n, 0, 500}], 105]][[1]]
CROSSREFS
Sequence in context: A019164 A108661 A117042 * A189088 A195301 A196824
KEYWORD
nonn,cons
AUTHOR
Michel Lagneau, Aug 02 2013
STATUS
approved