A295235 - OEIS

%I #33 Jul 23 2019 00:43:34

%S 0,1,2,3,4,5,6,7,8,9,10,12,14,15,16,17,18,20,21,24,28,30,31,32,33,34,

%T 36,40,42,48,56,60,62,63,64,65,66,68,72,73,80,84,85,96,112,120,124,

%U 126,127,128,129,130,132,136,144,146,160,168,170,192,224,240,248

%N Numbers k such that the positions of the ones in the binary representation of k are in arithmetic progression.

%C Also numbers k of the form Sum_{b=0..h-1} 2^(i+j*b) for some h >= 0, i >= 0, j > 0 (in fact, h = A000120(k), and if k > 0, i = A007814(k)).

%C There is a simple bijection between the finite sets of nonnegative integers in arithmetic progression and the terms of this sequence: s -> Sum_{i in s} 2^i; the term 0 corresponds to the empty set.

%C For any n > 0, A054519(n) gives the numbers of terms with n+1 digits in binary representation.

%C For any n >= 0, n is in the sequence iff 2*n is in the sequence.

%C For any n > 0, A000695(a(n)) is in the sequence.

%C The first prime numbers in the sequence are: 2, 3, 5, 7, 17, 31, 73, 127, 257, 8191, 65537, 131071, 262657, 524287, ...

%C This sequence contains the following sequences: A000051, A000079, A000225, A000668, A002450, A019434, A023001, A048645.

%C For any k > 0, 2^k - 2, 2^k - 1, 2^k, 2^k + 1 and 2^k + 2 are in the sequence (e.g., 14, 15, 16, 17, and 18).

%C Every odd term is a binary palindrome (and thus belongs to A006995).

%C Odd terms are A064896. - _Robert Israel_, Nov 20 2017

%H Rémy Sigrist, <a href="/A295235/b295235.txt">Table of n, a(n) for n = 1..1000</a>

%e The binary representation of the number 42 is "101010" and has ones evenly spaced, hence 42 appears in the sequence.

%e The first terms, alongside their binary representations, are:

%e    n  a(n)  a(n) in binary

%e   --  ----  --------------

%e    1    0           0

%e    2    1           1

%e    3    2          10

%e    4    3          11

%e    5    4         100

%e    6    5         101

%e    7    6         110

%e    8    7         111

%e    9    8        1000

%e   10    9        1001

%e   11   10        1010

%e   12   12        1100

%e   13   14        1110

%e   14   15        1111

%e   15   16       10000

%e   16   17       10001

%e   17   18       10010

%e   18   20       10100

%e   19   21       10101

%e   20   24       11000

%p f:= proc(d) local i,j,k;

%p   op(sort([seq(seq(add(2^(d-j*k),k=0..m),m=1..d/j),j=1..d),2^(d+1)]))

%p end proc:

%p 0,1,seq(f(d),d=0..10); # _Robert Israel_, Nov 20 2017

%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];

%t Select[Range[100],SameQ@@Differences[bpe[#]]&] (* _Gus Wiseman_, Jul 22 2019 *)

%o (PARI) is(n) = my(h=hammingweight(n)); if(h<3, return(1), my(i=valuation(n,2),w=#binary(n)); if((w-i-1)%(h-1)==0, my(j=(w-i-1)/(h-1)); return(sum(k=0,h-1,2^(i+j*k))==n), return(0)))

%Y Cf. A000051, A000079, A000120, A000225, A000668, A000695, A002450, A006995, A007814, A019434, A023001, A048645, A054519, A064896.

%Y Cf. A029931, A048793 (binary indices triangle), A070939, A291166, A325328 (prime indices rather than binary indices), A326669, A326675.

%K nonn,base

%O 1,3

%A _Rémy Sigrist_, Nov 18 2017