A272653 Numbers whose binary expansion is an abelian square.

3, 9, 10, 15, 33, 34, 36, 43, 45, 46, 51, 53, 54, 63, 129, 130, 132, 136, 147, 149, 150, 153, 154, 156, 163, 165, 166, 169, 170, 172, 183, 187, 189, 190, 195, 197, 198, 201, 202, 204, 215, 219, 221, 222, 231, 235, 237, 238, 255, 513, 514, 516, 520, 528, 547

Offset: 1

Author

N. J. A. Sloane, May 14 2016

Keywords

• nonn
• base

Comments

Numbers whose binary expansion has the form uv, where u begins with 1 and v is some permutation of u.

Could also be read as a table where row n gives the A178244(n) terms corresponding to u = (n written in binary), cf. Example section. - M. F. Hasler, Feb 23 2023

Examples

			34_10 = 100010_2 is a member, since v = 010 is a permutation of u = 100.
From _M. F. Hasler_, Feb 23 2023: (Start)
Grouping together in rows terms with the same u = binary(n):
   n |   u  | permutations v of u | decimal values of concat(u,v) read in binary
   1 |   1  |           1         | 3
   2 |  10  |        01, 10       | 9, 10
   3 |  11  |          11         | 15
   4 |  100 |    001, 010, 100    | 33, 34, 36
   5 |  101 |    011, 101, 110    | 43, 45, 46
   6 |  110 |         idem        | 51, 53, 54
   7 |  111 |         111         | 63
   8 | 1000 | 0001,0010,0100,1000 | 129, 130, 132, 136
   9 | 1001 | 0011, 0101, 0110,   | 147, 149, 150,
     |      |    1001, 1010, 1100 |    153, 154, 156
  ...|  ... | ...                 | ...
(End)
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A272654 (the binary expansions), A272655 (base 10 analog).

Programs

PARI

A272653_row(n, L=List())={forperm(vecsort(binary(n)), b, listput(L, n<<#b+fromdigits(Vec(b),2)));Vec(L)} \\ M. F. Hasler, Feb 23 2023

Python

from sympy.utilities.iterables import multiset_permutations
A272653_list = [int(b+''.join(s),2) for b in (bin(n)[2:] for n in range(1,100)) for s in multiset_permutations(sorted(b))] # Chai Wah Wu, May 15 2016

Extensions

More terms from Chai Wah Wu, May 15 2016