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 A295235 Numbers k such that the positions of the ones in the binary representation of k are in arithmetic progression. 15
 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 (list; graph; refs; listen; history; text; internal format)
 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, 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

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Last modified July 12 18:03 EDT 2020. Contains 335666 sequences. (Running on oeis4.)