A063511 a(n) = a(floor(sqrt(n))) * 2.

1, 2, 2, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8

Offset: 1

Author

Reinhard Zumkeller, Jul 30 2001

Keywords

• easy
• nonn

Comments

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)

Links

• Kevin Ryde, Table of n, a(n) for n = 1..8192
• Martin Fürer, Faster integer multiplication, Proceedings of the 39th Annual ACM Symposium on Theory of Computing, June 11-13 2007. And in SIAM Journal of Computing, volume 30, number 3, 2009, pages 979-1005. (See size "n" calculation at the start of Algorithm Integer-Multiplication.)
• Index to divisibility sequences

Crossrefs

Cf. A001146 (indices of new highs), A334789.

Programs

PARI

a(n) = if(n==1,1, 2<Kevin Ryde, May 11 2020

Formula

a(n) = 2^A211667(n) = 2^ceiling(log_2(log_2(n+1))). - Kevin Ryde, May 11 2020

log_2(n+1) <= a(n) < 2*log_2(n+1). - Charles R Greathouse IV, Nov 30 2024

Extensions

Formula and code by Charles R Greathouse IV moved to A334789 where they apply. - Kevin Ryde, May 11 2020