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A380872 Infinite square array, where row r >= 0 is the orbit of r under the map A380873: concatenate(sum of digits, product of digits).

%I A380872 #15 Apr 04 2025 22:37:13
%S A380872 0,0,1,0,11,2,0,21,22,3,0,32,44,33,4,0,56,816,69,44,5,0,1130,1548,
%T A380872 1554,816,55,6,0,50,18160,15100,1548,1025,66,7,0,50,160,70,18160,80,
%U A380872 1236,77,8,0,50,70,70,160,80,1236,1449,88,9,0,50,70,70,70,80,1236,18144,1664,99,10,0,50,70,70,70,80,1236,18128,17144,1881,10,11,0,50,70,70,70,80,1236,20128,17112,1864
%N A380872 Infinite square array, where row r >= 0 is the orbit of r under the map A380873: concatenate(sum of digits, product of digits).
%C A380872 As usual and required by the "table" display function, the array is read by falling antidiagonals.
%F A380872 A(r,0) = r; A(r,n+1) = A380873(A(r,n)) = concat(A007953(A(r,n)), A007954(A(r,n))).
%e A380872 The array starts as follows: (Elements in column 0 are also equal to the row index.)
%e A380872 col.0|  1 |  2 |  3  |  4  |  5  |  6 |  7 |  8  |  9  |  10 |  11 | 12 |  13 |  14
%e A380872 -----+----+----+-----+-----+-----+----+----+-----+-----+-----+-----+----+-----+-----
%e A380872    0    0     0     0     0     0    0    0     0     0     0     0    0     0     0
%e A380872    1   11    21    32    56  1130   50   50    50    50    50    50   50    50    50
%e A380872    2   22    44   816  1548 18160  160   70    70    70    70    70   70    70    70
%e A380872    3   33    69  1554 15100    70   70   70    70    70    70    70   70    70    70
%e A380872    4   44   816  1548 18160   160   70   70    70    70    70    70   70    70    70
%e A380872    5   55  1025    80    80    80   80   80    80    80    80    80   80    80    80
%e A380872    6   66  1236  1236  1236  1236 1236 1236  1236  1236  1236  1236 1236  1236  1236
%e A380872    7   77  1449 18144 18128 20128  130   40    40    40    40    40   40    40    40
%e A380872    8   88  1664 17144 17112  1214   88 1664 17144 17112  1214    88 1664 17144 17112
%e A380872    9   99  1881  1864 19192 22162 1348 1696 22324  1396 19162 19108  190   100    10
%e A380872   10   10    10    10    10    10   10   10    10    10    10    10   10    10    10
%e A380872   11   21    32    56  1130    50   50   50    50    50    50    50   50    50    50
%e A380872   ...  ...  ...
%e A380872 For example, row 1 is the trajectory of 1 under the map A380873: 1 -> concat (1,1) = 11 -> concat(1+1, 1*1) = 21 -> concat(2+1,2*1) = 32 -> concat(3+2,3*2) = 56 -> ...
%e A380872 Most of the  initial rows reach a fixed point after not too many iterations, but for example row 8 (A271268) and also 38, 83, 88, 146,... reach a cycle of length 5, C(88) = (88, 1664, 17144, 17112, 1214). Another 5-cycle is C(18168) = (18168, 24384, 21768, 24672, 21672), first reached in row 188 and 233.
%e A380872 Fixed points (see A062237) are the multiples of 10 less than 100, and 119 and 1236 (for row 6, 66, 123, ...), 19144 (row 289), and others.
%o A380872 (PARI) A380872_row(r, num_columns=30)=vector(num_columns, i, r=if(i>1, eval(Str(vecsum(r=digits(r)), if(r, vecprod(r)))), r))
%o A380872 A380872_array(rows=9, cols=rows)=Mat(vectorv(rows,i,A380872_row(i-1, cols)))
%Y A380872 Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits).
%Y A380872 Cf. A271220 (row 6), A271268 (row 8).
%K A380872 nonn,base,tabl
%O A380872 0,5
%A A380872 _M. F. Hasler_, Apr 01 2025