A272655 -id:A272655 - 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

A272654 Binary words beginning with 1 which are abelian squares.

Original entry on oeis.org

11, 1001, 1010, 1111, 100001, 100010, 100100, 101011, 101101, 101110, 110011, 110101, 110110, 111111, 10000001, 10000010, 10000100, 10001000, 10010011, 10010101, 10010110, 10011001, 10011010, 10011100, 10100011, 10100101, 10100110, 10101001, 10101010, 10101100

Offset: 1

N. J. A. Sloane, May 14 2016

Binary strings of the form uv, where u begins with 1 and v is some permutation of u.

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

Cf. A272653, A272655.

from sympy.utilities.iterables import multiset_permutations
A272654_list = [int(b+''.join(s)) 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

A384612 a(n) is the smallest integer k such that k^n is an abelian square; or -1 if no such k exists.

Original entry on oeis.org

11, 836, 11, 207, 624, 818222, 1001, 2776, 100001, 32323107, 100001, 85692627, 10000001, 501249084, 10000001, 27962757, 41695607, 70983559, 72768046, 977688137, 219873071, 112562383, 2338280974, 2435385853, 1231380445, 4557057314, 361499019, 8096434047, 5278552513

Offset: 1

Gonzalo Martínez, Jun 04 2025

Terms are the base of the smallest n-th power whose string of decimal digits is an abelian square; i.e., of the form m concatenated with a permutation of m (A272655).
If n is odd and A001700((n-1)/2) has d digits, then 0 < k <= 10^(2*d-1) + 1. - Robert Israel, Jun 05 2025
a(23) >= 1.83 * 10^9. Using Robert Israel's comment above a(23) <= 10^13 + 1. - David A. Corneth, Jun 06 2025

			a(1) = 11, since 11^1  = 1|1
a(2) = 836, since 836^2 = 698|896
a(3) = 11, since 11^3 = 13|31
a(4) = 207, since 207^4 = 18360|36801
a(5) = 624, since 624^5 = 9460692|9690624
a(6) = 818222, since 818222^6 = 300072996174564185|100579862765194304
a(7) = 1001, since 1001^7 = 10070210350|35021007001
a(8) = 2776, since 2776^8 = 35265958674713|24535718936576
a(9) = 100001, since 100001^9 = 10000900036000840012600|12600084000360000900001.

Chai Wah Wu, Table of n, a(n) for n = 1..34

Cf. A001597, A001700, A075786, A272655, A342942.

f:= proc(n) local k,d,x,L;
      k:= 2;
      do
        x:= k^n;
        d:= ilog10(x)+1;
        if d::odd then k:= ceil(10^(d/n)); next fi;
        L:= convert(x,base,10);
        if sort(L[1..d/2]) = sort(L[d/2+1..d]) then return k fi;
        k:= k+1
      od;
end proc:
map(f, [$1..30]); # Robert Israel, Jun 05 2025

from itertools import count
def ok(k, n):
    s = str(k**n)
    if len(s) % 2 != 0:
        return False
    mid = len(s) // 2
    return sorted(s[:mid]) == sorted(s[mid:])
def a(n):
    return next(k for k in count(2) if ok(k, n))
print([a(n) for n in range(1, 10)])

a(20)-a(22) from David A. Corneth, Jun 06 2025
a(23) from Gonzalo Martínez, Jun 06 2025
a(24)-a(29) from Jinyuan Wang, Jun 14 2025

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.

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A272653 Numbers whose binary expansion is an abelian square.

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A272654 Binary words beginning with 1 which are abelian squares.

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A384612 a(n) is the smallest integer k such that k^n is an abelian square; or -1 if no such k exists.

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

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Author

Keywords

Comments

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Crossrefs