A272653 - OEIS

A272653

Numbers whose binary expansion is an abelian square.

%I #19 Feb 26 2023 19:46:23

%S 3,9,10,15,33,34,36,43,45,46,51,53,54,63,129,130,132,136,147,149,150,

%T 153,154,156,163,165,166,169,170,172,183,187,189,190,195,197,198,201,

%U 202,204,215,219,221,222,231,235,237,238,255,513,514,516,520,528,547

%N Numbers whose binary expansion is an abelian square.

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

%C 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 Chai Wah Wu, <a href="/A272653/b272653.txt">Table of n, a(n) for n = 1..10000</a>

%e 34_10 = 100010_2 is a member, since v = 010 is a permutation of u = 100.

%e From _M. F. Hasler_, Feb 23 2023: (Start)

%e Grouping together in rows terms with the same u = binary(n):

%e n | u | permutations v of u | decimal values of concat(u,v) read in binary

%e 1 | 1 | 1 | 3

%e 2 | 10 | 01, 10 | 9, 10

%e 3 | 11 | 11 | 15

%e 4 | 100 | 001, 010, 100 | 33, 34, 36

%e 5 | 101 | 011, 101, 110 | 43, 45, 46

%e 6 | 110 | idem | 51, 53, 54

%e 7 | 111 | 111 | 63

%e 8 | 1000 | 0001,0010,0100,1000 | 129, 130, 132, 136

%e 9 | 1001 | 0011, 0101, 0110, | 147, 149, 150,

%e | | 1001, 1010, 1100 | 153, 154, 156

%e ...| ... | ... | ...

%e (End)

%o (Python)

%o from sympy.utilities.iterables import multiset_permutations

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

%K nonn,base

%O 1,1

%A _N. J. A. Sloane_, May 14 2016

%E More terms from _Chai Wah Wu_, May 15 2016