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, 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

Offset: 1

Author

Rémy Sigrist, Dec 17 2017

Keywords

• nonn
• base

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.

Examples

			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
		

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)

Crossrefs

Cf. A029837, A272679.