A272654 -id:A272654 - cp's OEIS frontend

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

N. J. A. Sloane, May 14 2016

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

			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)

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

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

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

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

More terms from Chai Wah Wu, May 15 2016

A272655 Numbers whose decimal expansion is an abelian square.

Original entry on oeis.org

11, 22, 33, 44, 55, 66, 77, 88, 99, 1001, 1010, 1111, 1212, 1221, 1313, 1331, 1414, 1441, 1515, 1551, 1616, 1661, 1717, 1771, 1818, 1881, 1919, 1991, 2002, 2020, 2112, 2121, 2222, 2323, 2332, 2424, 2442, 2525, 2552, 2626, 2662, 2727, 2772, 2828, 2882, 2929, 2992

Offset: 1

N. J. A. Sloane, May 14 2016

Decimal numbers of the form uv where the decimal digits of v are some permutation of the decimal digits of u.

			12344132 is a member because v = 4132 is a permutation of u = 1234.

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

Cf. A272653, A272654.

from sympy.utilities.iterables import multiset_permutations
A272655_list = [int(str(n)+''.join(s)) for n in range(1,100) for s in multiset_permutations(sorted(str(n)))] # Chai Wah Wu, May 15 2016

More terms from Chai Wah Wu, May 15 2016

A321252 (Conjecturally) the lexicographically earliest infinite sequence over {0,1,2,3} avoiding abelian squares.

Original entry on oeis.org

0, 1, 0, 2, 0, 1, 0, 3, 0, 1, 0, 2, 1, 0, 1, 3, 1, 0, 1, 2, 1, 3, 2, 0, 2, 1, 0, 1, 3, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 2, 0, 2, 3, 1, 0, 1, 2, 0, 2, 3, 2, 0, 2, 1, 2, 3, 0, 3, 2, 3, 0, 1, 0, 2, 0, 3, 0, 2, 0, 1, 2, 1, 3, 0, 1, 0, 2, 0, 3, 0, 1, 3, 0, 3, 2

Offset: 1

Jeffrey Shallit, Nov 01 2018

nonn

An abelian square is a nonempty string where the second half is a rearrangement of the first half, like the English word "reappear". To "avoid" an abelian square means to have no contiguous block of that form. Although an easy compactness argument, combined with a result of Keränen (1992) shows that the lexicographically earliest infinite string avoiding abelian squares over the alphabet {0,1,2,3} must exist, the terms provided are only conjecturally part of the described sequence, because we have no proof currently that this particular string can be extended infinitely far to the right.

Veikko Keränen, Abelian squares are avoidable on 4 letters, in ICALP 1992, volume 623 of Lecture Notes in Comput. Sci., pages 41-52. Springer-Verlag, 1992.

Cf. A272653, A272654, A272655.

cp's OEIS Frontend

A272653 Numbers whose binary expansion is an abelian square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

PARI

Python

Extensions

A272655 Numbers whose decimal expansion is an abelian square.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Python

Extensions

A321252 (Conjecturally) the lexicographically earliest infinite sequence over {0,1,2,3} avoiding abelian squares.

Views

Author

Keywords

Comments

Links

Crossrefs