Double Exponential Function - Doubly Exponential Sequences

Doubly Exponential Sequences

Aho and Sloane observed that in several important integer sequences, each term is a constant plus the square of the previous term, and show that such sequences can be formed by rounding to the nearest integer the values of a doubly exponential function in which the middle exponent is two. Integer sequences with this squaring behavior include

• The Fermat numbers

• The harmonic primes: The primes p, in which the sequence 1/2+1/3+1/5+1/7+....+1/p exceeds 0,1,2,3,....

The first few numbers, starting with 0, are 2,5,277,5195977,... (sequence A016088 in OEIS)

• The Double Mersenne numbers

• The elements of Sylvester's sequence (sequence A000058 in OEIS)

where E ≈ 1.264084735305302 is Vardi's constant.

• The number of k-ary operators:

More generally, if the nth value of an integer sequences is proportional to a double exponential function of n, Ionascu and Stanica call the sequence "almost doubly-exponential" and describe conditions under which it can be defined as the floor of a doubly-exponential sequence plus a constant. Additional sequences of this type include

• The prime numbers 2, 11, 1361, ... (sequence A051254 in OEIS)

where A ≈ 1.306377883863 is Mills' constant.