Egyptian fractions

Following the ancient Egyptian number system, rational numbers can be represented by sums of reciprocals, as in 3/7==1/3+1/11+1/231. With suitable distinct integers a[n] one can represent any number by Sum[1/a[n], {n, ∞}]. The representation is not unique; a[n]=2^{n}, n (n+1) and (n+1)!/n all yield 1. Simple choices for a[n] yield many standard transcendental numbers: n!: E-1; n!^{2}: BesselI[0,2]-1; n 2^{n}: Log[2]; n^{2}: Pi^{2}/6; (3n-1)(3n-2): Pi Sqrt[3]/9; 3-16n+16n^{2}: Pi/8; n n!: ExpIntegralEi[1] - EulerGamma. (See also page 902.)