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A063511 a(n) = a(floor(square root(n))) * 2. 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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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

FORMULA

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

PROG

(PARI) a(n) = if(n==1, 1, 2<<logint(logint(n, 2), 2)); \\ Kevin Ryde, May 11 2020

CROSSREFS

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

Sequence in context: A179932 A267649 A071805 * A334789 A283207 A164717

Adjacent sequences:  A063508 A063509 A063510 * A063512 A063513 A063514

KEYWORD

easy,nonn

AUTHOR

Reinhard Zumkeller, Jul 30 2001

EXTENSIONS

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

STATUS

approved

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Last modified February 28 05:24 EST 2021. Contains 341695 sequences. (Running on oeis4.)