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A272653 Numbers whose binary expansion is an abelian square.

%I A272653 #19 Feb 26 2023 19:46:23
%S A272653 3,9,10,15,33,34,36,43,45,46,51,53,54,63,129,130,132,136,147,149,150,
%T A272653 153,154,156,163,165,166,169,170,172,183,187,189,190,195,197,198,201,
%U A272653 202,204,215,219,221,222,231,235,237,238,255,513,514,516,520,528,547
%N A272653 Numbers whose binary expansion is an abelian square.
%C A272653 Numbers whose binary expansion has the form uv, where u begins with 1 and v is some permutation of u.
%C A272653 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
%H A272653 Chai Wah Wu, <a href="/A272653/b272653.txt">Table of n, a(n) for n = 1..10000</a>
%e A272653 34_10 = 100010_2 is a member, since v = 010 is a permutation of u = 100.
%e A272653 From _M. F. Hasler_, Feb 23 2023: (Start)
%e A272653 Grouping together in rows terms with the same u = binary(n):
%e A272653    n |   u  | permutations v of u | decimal values of concat(u,v) read in binary
%e A272653    1 |   1  |           1         | 3
%e A272653    2 |  10  |        01, 10       | 9, 10
%e A272653    3 |  11  |          11         | 15
%e A272653    4 |  100 |    001, 010, 100    | 33, 34, 36
%e A272653    5 |  101 |    011, 101, 110    | 43, 45, 46
%e A272653    6 |  110 |         idem        | 51, 53, 54
%e A272653    7 |  111 |         111         | 63
%e A272653    8 | 1000 | 0001,0010,0100,1000 | 129, 130, 132, 136
%e A272653    9 | 1001 | 0011, 0101, 0110,   | 147, 149, 150,
%e A272653      |      |    1001, 1010, 1100 |    153, 154, 156
%e A272653   ...|  ... | ...                 | ...
%e A272653 (End)
%o A272653 (Python)
%o A272653 from sympy.utilities.iterables import multiset_permutations
%o A272653 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
%o A272653 (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
%Y A272653 Cf. A272654 (the binary expansions), A272655 (base 10 analog).
%K A272653 nonn,base
%O A272653 1,1
%A A272653 _N. J. A. Sloane_, May 14 2016
%E A272653 More terms from _Chai Wah Wu_, May 15 2016