A230097 Indices of records in A159918.

0, 1, 3, 5, 11, 21, 39, 45, 75, 155, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 908091, 980853, 2960011, 2965685, 5931189, 11862197, 20437147, 22193965, 43586515, 57804981, 157355851

Offset: 1

Author

N. J. A. Sloane, Oct 11 2013

Keywords

• nonn
• base

Comments

The records themselves are not so interesting: 0, 1, 2, 3, 5, 6, 7, 8, 9, 10, 13, 14, 15, 16, 17, 18, 19, 20, ... (A357304).

Lindström mentions that the record value 34 in A159918 is first reached at n = 980853.

Links

• Bert Dobbelaere, Table of n, a(n) for n = 1..80, (terms 41..64 from Donovan Johnson, 65..70 from Hugo Pfoertner, missing 68 and 72..80 from Bert Dobbelaere).
• Bernt Lindström, On the binary digits of a power, Journal of Number Theory, Volume 65, Issue 2, August 1997, Pages 321-324.

Crossrefs

Cf. A000120, A159918, A231897, A357304, A357658.

Programs

Haskell

a230097 n = a230097_list !! (n-1)
a230097_list = 0 : f 0 0 where
   f i m = if v > m then i : f (i + 1) v else f (i + 1) m
           where v = a159918 i
-- Reinhard Zumkeller, Oct 12 2013
(Python 3.10+)
from itertools import count, islice
def A230097_gen(): # generator of terms
    c = -1
    for n in count(0):
        if (m := (n**2).bit_count())>c:
            yield n
            c = m
A230097_list = list(islice(A230097_gen(),20)) # Chai Wah Wu, Oct 01 2022

Formula

Lindström shows that lim sup wt(m^2)/log_2 m = 2.

Extensions

a(19)-a(40) from Reinhard Zumkeller, Oct 12 2013