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A296616 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, the binary expansion of a(n) * a(n + 1) starts with the binary expansion of n. 1
1, 2, 4, 3, 6, 7, 14, 8, 16, 9, 18, 5, 10, 11, 21, 12, 22, 13, 23, 27, 24, 28, 26, 29, 53, 31, 54, 32, 56, 17, 57, 35, 15, 36, 61, 37, 63, 19, 64, 39, 33, 20, 34, 41, 69, 42, 71, 43, 72, 44, 73, 45, 74, 46, 38, 47, 77, 48, 78, 49, 79, 25, 40, 51, 81, 52, 82 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

It is likely that this sequence is a permutation of the natural numbers.

The lines visible in the scatterplot of the first terms seems to corresponds to set of indices n where the function f(n) = Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)) has the same value; those lines can be partitioned into two groups, depending on the parity of n (see Links section).

This sequence has connections with A272679: here the binary expansion of a(n)*a(n+1) starts with that of n, there the binary expansion of a(n)^2 starts with that of n.

LINKS

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Rémy Sigrist, C++ program for A296616

Rémy Sigrist, Colored scatterplot of the first 100000 terms (where the color is function of Sum_{k=1..n-1} (-1)^k * (A029837(1+a(k)*a(k+1)) - A029837(1+k)))

Rémy Sigrist, Colored scatterplot of the first 10000 terms (where the color is function of the parity of n)

EXAMPLE

The first terms, alongside the binary representations of n and a(n) * a(n + 1), are:

  n     a(n)    bin(n)    bin(a(n)*a(n+1))

  --    ----    ------    ----------------

   1       1         1            10

   2       2        10          1000

   3       4        11          1100

   4       3       100         10010

   5       6       101        101010

   6       7       110       1100010

   7      14       111       1110000

   8       8      1000      10000000

   9      16      1001      10010000

  10       9      1010      10100010

  11      18      1011       1011010

  12       5      1100        110010

  13      10      1101       1101110

  14      11      1110      11100111

  15      21      1111      11111100

  16      12     10000     100001000

  17      22     10001     100011110

  18      13     10010     100101011

  19      23     10011    1001101101

  20      27     10100    1010001000

PROG

(C++) See Links section.

CROSSREFS

Cf. A029837, A272679.

Sequence in context: A048680 A342793 A178898 * A143529 A331010 A261351

Adjacent sequences:  A296613 A296614 A296615 * A296617 A296618 A296619

KEYWORD

nonn,base

AUTHOR

Rémy Sigrist, Dec 17 2017

STATUS

approved

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Last modified June 12 20:08 EDT 2021. Contains 344963 sequences. (Running on oeis4.)