cp's OEIS Frontend

From Kevin Ryde, May 11 2020: (Start)

The sqrt steps in the definition are equivalent to A211667 but here factors of 2 instead of counting, so a(n) = 2^A211667(n). A211667 is a double logarithm and the effect of power 2^ is to turn the second into a rounding. So a(n) is the bit length of n (see A070939) increased to the next power of 2 if not already a power of 2. Each n = 2^(2^k) is a new high a(n) = 2^(k+1), since such an n is bit length 2^k+1.

In a microcomputer, it's common for machine words to be power-of-2 sizes such as 16, 32, 64, 128 bits. a(n) can be thought of as the word size needed to contain integer n. Some algorithms by their nature expect power-of-2 sizes, for example Schönhage and Strassen's big integer multiplication.

This sequence differs from A334789 (2^log*(n)) for n>=256. For example a(256)=16 whereas A334789(256)=8. The respective exponent sequences are A211667 (for here) and A001069 (for A334789) which likewise differ for n>=256.

(End)

Differs from A063511 for n>=256. For example a(256)=8 whereas A063511(256)=16. The respective exponent sequences are A001069 (for here) and A211667 (for A063511) which likewise differ for n>=256.

2^log*(n) arises in computational complexity measures for Fürer's multiplication algorithm.