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

1,2

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

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

Formula

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

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

Example

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

Prog

(PARI) See Links section.

Crossrefs

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

Sequence in context: A342045 A307591 A031275 * A276104 A376370 A329804

Adjacent sequences: A306462 A306463 A306464 * A306466 A306467 A306468

Keyword

nonn,base

Author

Rémy Sigrist, Feb 17 2019

Status

approved