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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.

%I A306465 #17 Feb 19 2019 10:34:04
%S A306465 1,2,3,10,4,11,5,100,6,101,7,110,8,111,9,1000,12,13,20,14,21,22,30,23,
%T A306465 102,24,112,31,32,103,33,120,40,121,41,200,34,201,42,202,43,1001,15,
%U A306465 1010,16,1011,17,1100,18,1101,25,1110,26,1111,27,10000,19,10001,28
%N 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.
%C A306465 This sequence is the variant of A266195 in base 10.
%C A306465 This sequence is a permutation of the natural numbers, with inverse A306466. Proof:
%C A306465 - we can always extend the sequence with a power of ten not yet in the sequence, hence the sequence is well defined and infinite,
%C A306465 - for any k > 0, 10^(k-1) is the first k-digit number appearing in the sequence,
%C A306465 - all powers of ten appear in the sequence, in increasing order,
%C A306465 - a power of ten is always followed by the least number unused so far,
%C A306465 hence every number eventually appears. QED
%H A306465 Rémy Sigrist, <a href="/A306465/b306465.txt">Table of n, a(n) for n = 1..10000</a>
%H A306465 Rémy Sigrist, <a href="/A306465/a306465.gp.txt">PARI program for A306465</a>
%H A306465 Rémy Sigrist, <a href="/A306465/a306465.png">Colored logarithmic scatterplot of the sequence for n = 1..200000</a> (where the color is function of A054055(a(n)))
%H A306465 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A306465 A007953(a(n) * a(n+1)) = A007953(a(n)) * A007953(a(n+1)).
%F A306465 A054055(a(n)) * A054055(a(n+1)) <= 9.
%e A306465 The first terms, alongside their digital sum and the digital sum of the product with the next term, are:
%e A306465   n   a(n)  ds(a(n))  ds(a(n)*a(n+1))
%e A306465   --  ----  --------  ---------------
%e A306465    1     1         1                2
%e A306465    2     2         2                6
%e A306465    3     3         3                3
%e A306465    4    10         1                4
%e A306465    5     4         4                8
%e A306465    6    11         2               10
%e A306465    7     5         5                5
%e A306465    8   100         1                6
%e A306465    9     6         6               12
%e A306465   10   101         2               14
%e A306465   11     7         7               14
%e A306465   12   110         2               16
%e A306465   13     8         8               24
%e A306465   14   111         3               27
%e A306465   15     9         9                9
%e A306465   16  1000         1                3
%e A306465   17    12         3               12
%o A306465 (PARI) See Links section.
%Y A306465 Cf. A007953, A054055, A266195, A306466 (inverse).
%K A306465 nonn,base
%O A306465 1,2
%A A306465 _Rémy Sigrist_, Feb 17 2019