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Decimal expansion of Sum_{n>=1} d(n)/n!, where d(n) is the number of divisors of n.
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%I #7 Mar 12 2024 02:48:45

%S 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,

%T 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,

%U 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

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

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

%H Paul Erdős and Ernst G. Straus, <a href="http://dx.doi.org/10.2140/pjm.1971.36.635">Some number theoretic results</a>, Pacific Journal of Mathematics, Vol. 36, No. 3 (1971), pp. 635-646.

%H Michael Ian Shamos, <a href="http://euro.ecom.cmu.edu/people/faculty/mshamos/Overcounting.pdf">Overcounting Functions</a>, 2011.

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

%e 2.48106101979076269793744769639865739568689776121713...

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

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

%o (PARI) suminf(k=1,numdiv(k)/k!)

%Y Cf. A000005, A336334.

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

%K nonn,cons

%O 1,1

%A _Amiram Eldar_, Mar 12 2024