A306465 - OEIS

%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