A295235 Numbers k such that the positions of the ones in the binary representation of k are in arithmetic progression.

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, 36, 40, 42, 48, 56, 60, 62, 63, 64, 65, 66, 68, 72, 73, 80, 84, 85, 96, 112, 120, 124, 126, 127, 128, 129, 130, 132, 136, 144, 146, 160, 168, 170, 192, 224, 240, 248

Offset

1,3

Comments

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)).

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.

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

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

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

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

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

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).

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

Odd terms are A064896. - Robert Israel, Nov 20 2017

Links

Rémy Sigrist, Table of n, a(n) for n = 1..1000

Example

The binary representation of the number 42 is "101010" and has ones evenly spaced, hence 42 appears in the sequence.
The first terms, alongside their binary representations, are:
   n  a(n)  a(n) in binary
  --  ----  --------------
   1    0           0
   2    1           1
   3    2          10
   4    3          11
   5    4         100
   6    5         101
   7    6         110
   8    7         111
   9    8        1000
  10    9        1001
  11   10        1010
  12   12        1100
  13   14        1110
  14   15        1111
  15   16       10000
  16   17       10001
  17   18       10010
  18   20       10100
  19   21       10101
  20   24       11000

Maple

f:= proc(d) local i, j, k;
  op(sort([seq(seq(add(2^(d-j*k), k=0..m), m=1..d/j), j=1..d), 2^(d+1)]))
end proc:
0, 1, seq(f(d), d=0..10); # Robert Israel, Nov 20 2017

Mathematica

bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];
Select[Range[100], SameQ@@Differences[bpe[#]]&] (* Gus Wiseman, Jul 22 2019 *)

Prog

(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)))

Crossrefs

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

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

Sequence in context: A161604 A125121 A333762 * A136490 A129523 A243463

Adjacent sequences: A295232 A295233 A295234 * A295236 A295237 A295238

Keyword

nonn,base

Author

Rémy Sigrist, Nov 18 2017

Status

approved