A302659 Lexicographically earliest sequence of distinct positive terms such that, for any n > 0, A065368(a(n) * a(n+1)) = 0.

1, 4, 2, 6, 8, 3, 12, 7, 16, 5, 24, 9, 28, 10, 14, 18, 20, 11, 32, 15, 44, 13, 36, 17, 40, 19, 52, 21, 48, 22, 26, 30, 34, 38, 42, 46, 50, 56, 23, 60, 25, 64, 27, 68, 29, 72, 31, 76, 35, 80, 33, 84, 37, 88, 41, 96, 43, 92, 39, 100, 45, 112, 47, 116, 49, 104

Offset: 1

Author

Rémy Sigrist, Apr 11 2018

Keywords

• nonn
• base

Comments

For any base b > 1:

- let s_b be the alternate sum of digits in base b,

- in particular s_2 = A065359, s_3 = A065368 and s_10 = A225693,

- we can build an analog of this sequence, say f_b, by considering s_b instead of A065368,

- f_b is well defined: for any k > 0, s_b(k * (1 + b^(2*i + 1))) = 0 whenever k < b^(2*i + 1), hence we can always extend the sequence,

- f_b is conjectured to be a permutation of the natural numbers,

- the scatterplot of f_b shows lines that seem related to the value of s_b(f_b) mod b+1.

See A302544 for a similar sequence.

Examples

			The first terms, alongside the ternary representation of a(n) * a(n+1), are:
  n  a(n)    tern(a(n) * a(n+1))
  -- ----    -------------------
   1    1        11
   2    4        22
   3    2       110
   4    6      1210
   5    8       220
   6    3      1100
   7   12     10010
   8    7     11011
   9   16      2222
  10    5     11110
  11   24     22000
  12    9    100100
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, Colored scatterplot of the first 10000 terms
• Rémy Sigrist, Scatterplot of the first 10000 of f_2
• Rémy Sigrist, Scatterplot of the first 10000 of f_5
• Rémy Sigrist, PARI program for A302659
• Index entries for sequences related to digital root, sum, etc.

Crossrefs

Cf. A065359, A065368, A225693, A302544.

Programs

PARI

See Links section.