|
%I #54 Jul 28 2019 21:46:24
%S 1,2,3,4,8,10,15,16,32,64,128,136,170,255,256,512,1024,2048,4096,8192,
%T 16384,32768,32896,34952,43690,65535,65536,131072,262144,524288,
%U 1048576,2097152,4194304,8388608,16777216,33554432,67108864,134217728
%N Numbers k such that the binary plot of the list of divisors of k has reflection symmetry.
%C The sequence is infinite as it contains every power of 2 (A000079).
%C The product of the first five Fermat primes (A019434), 4294967295 = 3 * 5 * 17 * 257 * 65537, is also a member of this sequence.
%C Every term belongs to A135772.
%C The first 48 terms are all of the form Sum_{i=1..t} 2^(k*t-1) for some k > 0 and t > 0 (see binary plot in Links section).
%H Rémy Sigrist, <a href="/A308811/a308811.png">Binary plot of the first 48 terms</a>
%F A295368(a(n)) = a(n).
%e Regarding 170:
%e - the divisors of 170 are: 1, 2, 5, 10, 17, 34, 85, 170,
%e - in binary: "1", "10", "101", "1010", "10001", "100010", "1010101", "10101010",
%e - the corresponding binary plot is:
%e . 1 . #
%e . 1 0 . #
%e . 1 0 1 . # #
%e . 1 0 1 0 . # #
%e 1 0 0 0 1 # . #
%e 1 0 0 0 1 0 # . #
%e 1 0 1 0 1 0 1 # # # . #
%e 1 0 1 0 1 0 1 0 # # # # .
%e . .
%e . .
%e - this binary plot has reflection symmetry,
%e - hence 170 belongs to this sequence.
%o (PARI) is(n) = { my (d=Vecrev(divisors(n))); if (#binary(d[1])==#d, for (b=0, #d-1, my (t=0); for (i=1, #d, if (bittest(d[i], b), t+=2^(i-1))); if (t!=d[b+1], return (0))); return (1), return (0)) }
%Y Cf. A000079, A019434, A135772, A295368.
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Jul 08 2019
|
