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

Offset

1,1

Comments

Problem No. 45 from P. Erdős (see the first reference). 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 the second reference for the proof).

Links

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

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

Cf. A000203, A227989.

Sequence in context: A113966 A164611 A316086 * A182813 A334627 A073897

Adjacent sequences: A227985 A227986 A227987 * A227989 A227990 A227991

Keyword

nonn,cons

Author

Michel Lagneau, Aug 02 2013

Status

approved