A227989 -id:A227989 - cp's OEIS frontend

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

Michel Lagneau, Aug 02 2013

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).

			3.52700047185295282976153...

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

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.

Cf. A000203, A227989.

with(numtheory):Digits:=200: s:=evalf(sum('sigma(i)/i!', 'i'=1..500)):print(s):

RealDigits[N[Sum[DivisorSigma[1,n]/n!, {n, 0, 500}], 200]][[1]]

suminf(n=1, sigma(n)/n!) \\ Michel Marcus, Sep 16 2017

A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Jun 21 2020

			7.099285178890907114033125022164753663157608833211895...

Maxie Dion Schmidt, A catalog of interesting and useful Lambert series identities, arXiv:2004.02976 [math.NT], 2020.
Eric Weisstein's World of Mathematics, Lambert Series.
Wikipedia, Lambert series.

Cf. A001157, A065442, A066766, A227989, A256936, A335764.

evalf(add( (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3, n = 1..20 ), 100); # Peter Bala, Jan 22 2021

RealDigits[Sum[n^2/(2^n - 1), {n, 1, 500}], 10, 100][[1]]

Equals Sum_{k>=1} k^2/(2^k - 1).

Faster converging series: Sum_{n >= 1} (1/2)^(n^2)*( n^2*(8^n) - ((n-1)^2 - 2)*4^n - ((n+1)^2 - 2)*(2^n) + n^2 )/(2^n - 1)^3. - Peter Bala, Jan 19 2021

A307036 Decimal expansion of Sum_{k >= 1} sigma_3(k)/k!, where sigma_3(k) is the sum of cubes of the divisors of k (A001158).

Original entry on oeis.org

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

Offset: 2

Amiram Eldar, Mar 21 2019

Its irrationality was conjectured by Erdős and Kac in 1953 and was proved by Schlage-Puchta in 2006 and Friedlander et al. in 2007.

			14.6935328472692822331235513649820566388631939576...

Paul Erdős and Mark Kac, Problem 4518, American Mathematical Monthly, Vol. 60, No. 1 (1953), p. 47.
John B. Friedlander, Florian Luca, and Mihai Stoiciu, On the irrationality of a divisor function series, Integers, Vol. 7, No. 1 (2007), A31.
Jan-Christoph Schlage-Puchta, The irrationality of a number theoretical series, The Ramanujan Journal, Vol. 12, No. 3 (2006), pp. 455-460, preprint, arXiv:1105.1452 [math.NT], 2011.

Cf. A001158, A227988, A227989.

RealDigits[N[Sum[DivisorSigma[3, n]/n!, {n, 1, 500}], 100]][[1]]

suminf(k=1, sigma(k, 3)/k!) \\ Michel Marcus, Mar 21 2019

A359060 Decimal expansion of Sum_{n >= 1} sigma_4(n)/n!.

Original entry on oeis.org

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

Offset: 2

Charles R Greathouse IV, Dec 14 2022

This constant's irrationality was conjectured by Erdős and Kac in 1953 and proved by Pratt in 2022.

			42.301047503733508066864284062530764530595670622493323155118876949426899131....

Paul Erdős and Mark Kac, Problem 4518, American Mathematical Monthly, Vol. 60, No. 1 (1953), p. 47.
Kyle Pratt, The irrationality of a divisor function series of Erdős and Kac, arXiv preprint, arXiv:2209.11124 [math.NT], 2022.

Sum_{n >= 1} sigma_k(n)/n!: A227988 (k=1), A227989 (k=2), A307036 (k=3), this sequence (k=4).

RealDigits[N[Sum[DivisorSigma[4, n]/n!, {n, 1, 500}], 120]][[1]] (* Amiram Eldar, Jun 21 2023 *)

PARI
```
suminf(n=1,sigma(n,4)/n!)
```

A371133 Decimal expansion of Sum_{n>=1} d(n)/n!, where d(n) is the number of divisors of n.

Original entry on oeis.org

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

Offset: 1

Amiram Eldar, Mar 12 2024

This constant is irrational (Erdős and Straus, 1971).

			2.48106101979076269793744769639865739568689776121713...

Paul Erdős and Ernst G. Straus, Some number theoretic results, Pacific Journal of Mathematics, Vol. 36, No. 3 (1971), pp. 635-646.
Michael Ian Shamos, Overcounting Functions, 2011.

Cf. A000005, A336334.

Sum_{n>=1} sigma_k(n)/n!: this sequence (k=0), A227988 (k=1), A227989 (k=2), A307036 (k=3), A359060 (k=4).

with(numtheory); evalf(Sum(tau(n)/factorial(n), n = 1 .. infinity), 120)

RealDigits[N[Sum[DivisorSigma[0, n]/n!, {n, 1, 500}], 120]][[1]]

PARI
```
suminf(k=1,numdiv(k)/k!)
```

Equals Sum_{j,k>=1} 1/(j*k)! (Shamos, 2011, p. 4).

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A227988 Decimal expansion of Sum_{n >= 1} sigma_1(n)/n!.

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A335763 Decimal expansion of Sum_{k>=1} sigma_2(k)/2^k where sigma_2(k) is the sum of squares of divisors of k (A001157).

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A307036 Decimal expansion of Sum_{k >= 1} sigma_3(k)/k!, where sigma_3(k) is the sum of cubes of the divisors of k (A001158).

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A359060 Decimal expansion of Sum_{n >= 1} sigma_4(n)/n!.

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A371133 Decimal expansion of Sum_{n>=1} d(n)/n!, where d(n) is the number of divisors of n.

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