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

1,1

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

Links

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

Example

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)

Prog

(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
(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

Crossrefs

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

Sequence in context: A324584 A325284 A049096 * A030794 A134073 A088005

Adjacent sequences: A272650 A272651 A272652 * A272654 A272655 A272656

Keyword

nonn,base

Author

N. J. A. Sloane, May 14 2016

Extensions

More terms from Chai Wah Wu, May 15 2016

Status

approved