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%I #17 Feb 19 2019 10:34:04
%S 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 102,24,112,31,32,103,33,120,40,121,41,200,34,201,42,202,43,1001,15,
%U 1010,16,1011,17,1100,18,1101,25,1110,26,1111,27,10000,19,10001,28
%N 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 This sequence is the variant of A266195 in base 10.
%C This sequence is a permutation of the natural numbers, with inverse A306466. Proof:
%C - 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 - for any k > 0, 10^(k-1) is the first k-digit number appearing in the sequence,
%C - all powers of ten appear in the sequence, in increasing order,
%C - a power of ten is always followed by the least number unused so far,
%C hence every number eventually appears. QED
%H Rémy Sigrist, <a href="/A306465/b306465.txt">Table of n, a(n) for n = 1..10000</a>
%H Rémy Sigrist, <a href="/A306465/a306465.gp.txt">PARI program for A306465</a>
%H 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 <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%F A007953(a(n) * a(n+1)) = A007953(a(n)) * A007953(a(n+1)).
%F A054055(a(n)) * A054055(a(n+1)) <= 9.
%e The first terms, alongside their digital sum and the digital sum of the product with the next term, are:
%e n a(n) ds(a(n)) ds(a(n)*a(n+1))
%e -- ---- -------- ---------------
%e 1 1 1 2
%e 2 2 2 6
%e 3 3 3 3
%e 4 10 1 4
%e 5 4 4 8
%e 6 11 2 10
%e 7 5 5 5
%e 8 100 1 6
%e 9 6 6 12
%e 10 101 2 14
%e 11 7 7 14
%e 12 110 2 16
%e 13 8 8 24
%e 14 111 3 27
%e 15 9 9 9
%e 16 1000 1 3
%e 17 12 3 12
%o (PARI) See Links section.
%Y Cf. A007953, A054055, A266195, A306466 (inverse).
%K nonn,base
%O 1,2
%A _Rémy Sigrist_, Feb 17 2019