A306465 Lexicographically earliest sequence of distinct positive terms such that the product of any two consecutive terms can be computed without carry by long multiplication in base 10.

1, 2, 3, 10, 4, 11, 5, 100, 6, 101, 7, 110, 8, 111, 9, 1000, 12, 13, 20, 14, 21, 22, 30, 23, 102, 24, 112, 31, 32, 103, 33, 120, 40, 121, 41, 200, 34, 201, 42, 202, 43, 1001, 15, 1010, 16, 1011, 17, 1100, 18, 1101, 25, 1110, 26, 1111, 27, 10000, 19, 10001, 28

Offset: 1

Author

Rémy Sigrist, Feb 17 2019

Keywords

• nonn
• base

Comments

This sequence is the variant of A266195 in base 10.

This sequence is a permutation of the natural numbers, with inverse A306466. Proof:

- we can always extend the sequence with a power of ten not yet in the sequence, hence the sequence is well defined and infinite,

- for any k > 0, 10^(k-1) is the first k-digit number appearing in the sequence,

- all powers of ten appear in the sequence, in increasing order,

- a power of ten is always followed by the least number unused so far,

hence every number eventually appears. QED

Examples

			The first terms, alongside their digital sum and the digital sum of the product with the next term, are:
  n   a(n)  ds(a(n))  ds(a(n)*a(n+1))
  --  ----  --------  ---------------
   1     1         1                2
   2     2         2                6
   3     3         3                3
   4    10         1                4
   5     4         4                8
   6    11         2               10
   7     5         5                5
   8   100         1                6
   9     6         6               12
  10   101         2               14
  11     7         7               14
  12   110         2               16
  13     8         8               24
  14   111         3               27
  15     9         9                9
  16  1000         1                3
  17    12         3               12
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10000
• Rémy Sigrist, PARI program for A306465
• Rémy Sigrist, Colored logarithmic scatterplot of the sequence for n = 1..200000 (where the color is function of A054055(a(n)))
• Index entries for sequences that are permutations of the natural numbers

Crossrefs

Cf. A007953, A054055, A266195, A306466 (inverse).

Programs

PARI

See Links section.

Formula

A007953(a(n) * a(n+1)) = A007953(a(n)) * A007953(a(n+1)).

A054055(a(n)) * A054055(a(n+1)) <= 9.