A227988 - OEIS

A227988

Decimal expansion of Sum_{n >= 1} sigma_1(n)/n!.

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

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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

Table of n, a(n) for n=1..87.

Thomas Bloom, Problem 252, Erdős Problems.

Paul Erdős, Some unsolved problems, Publ. Inst. Hung. Acad. Sci. 6 (1961), 221-259.

Paul 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.

Paul Erdős, On the irrationality of certain series: problems and results, in New advances in Transcendence Theory, Cambridge Univ. Press, 1988, pp. 102-109.

Paul Erdős and M. Kac, Problem 4518, Amer. Math. Monthly 60 (1953) 47. Solution R. Breusch, 61 (1954) 264-265.

Erdős problems database contributors, Erdős problem database, see no. 252.

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

Cf. A000203, A227989.