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

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

Author

Amiram Eldar, Mar 12 2024

Keywords

• nonn
• cons

Comments

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

Examples

			2.48106101979076269793744769639865739568689776121713...
		

Links

• 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.

Crossrefs

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).

Programs

Maple

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

Mathematica

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

PARI

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

Formula

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