A231897 a(n) = smallest m such that wt(m^2) = n (where wt(i) = A000120(i)), or -1 if no such m exists.

0, 1, 3, 5, 13, 11, 21, 39, 45, 75, 155, 217, 331, 181, 627, 923, 1241, 2505, 3915, 5221, 6475, 11309, 15595, 19637, 31595, 44491, 69451, 113447, 185269, 244661, 357081, 453677, 1015143, 908091, 980853, 2960011, 4568757, 2965685, 5931189, 11862197, 20437147

Offset: 0

Author

N. J. A. Sloane, Nov 19 2013

Keywords

• nonn
• base

Comments

Conjecture: a(n) is never -1. (It seems likely that the arguments of Lindström (1997) could be modified to establish this conjecture.)

a(n) is the smallest m such that A159918(m) = n (or -1 if ...).

Links

• Hugo Pfoertner, Table of n, a(n) for n = 0..110 (terms 0..70 from Donovan Johnson, significant extension enabled by programs provided in Code Golf challenge).
• Code Golf Stackexchange, Smallest and largest 100-bit square with maximum Hamming weight, fastest code challenge started Dec 15 2022.
• 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, A230097, A211201, A231898, A214560.

A089998 are the corresponding squares.

Programs

Haskell

a231897 n = head [x | x <- [1..], a159918 x == n]
-- Reinhard Zumkeller, Nov 20 2013

PARI

a(n)=if(n,my(k); while(hammingweight(k++^2)!=n,); k, 0) \\ Charles R Greathouse IV, Aug 06 2015

Python

def wt(n): return bin(n).count('1')
def a(n):
    m = 2**(n//2) - 1
    while wt(m**2) != n: m += 1
    return m
print([a(n) for n in range(32)]) # Michael S. Branicky, Feb 06 2022

Formula

a(n) = 2*A211201(n-1) + 1 for n >= 1. - Hugo Pfoertner, Feb 06 2022

Extensions

a(26)-a(40) from Reinhard Zumkeller, Nov 20 2013