A307036 -id:A307036 - cp's OEIS frontend

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

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).

cp's OEIS Frontend

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

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