OFFSET
1,1
COMMENTS
Problem No. 45 from P. Erdős (see the 1963 link). 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 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, Geneva, 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
3.52700047185295282976153...
MAPLE
with(numtheory):Digits:=200: s:=evalf(sum('sigma(i)/i!', 'i'=1..500)):print(s):
MATHEMATICA
RealDigits[N[Sum[DivisorSigma[1, n]/n!, {n, 0, 500}], 200]][[1]]
PROG
(PARI) suminf(n=1, sigma(n)/n!) \\ Michel Marcus, Sep 16 2017
CROSSREFS
KEYWORD
nonn,cons
AUTHOR
Michel Lagneau, Aug 02 2013
STATUS
approved