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] = 2ⁿ, n (n + 1) and (n + 1)!/n all yield 1. Simple choices for a[n] yield many standard transcendental numbers: n!:  - 1; n!²: BesselI[0, 2] - 1; n 2ⁿ: Log[2]; n²: π²/6; (3n - 1)(3n - 2): π √3/9; 3 - 16n + 16n²: π/8; n n!: ExpIntegralEi[1] - EulerGamma. (See also page 902.)