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

0, 0, 1, 0, 11, 2, 0, 21, 22, 3, 0, 32, 44, 33, 4, 0, 56, 816, 69, 44, 5, 0, 1130, 1548, 1554, 816, 55, 6, 0, 50, 18160, 15100, 1548, 1025, 66, 7, 0, 50, 160, 70, 18160, 80, 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

Offset: 0

Author

M. F. Hasler, Apr 01 2025

Keywords

• nonn
• base
• tabl

Comments

As usual and required by the "table" display function, the array is read by falling antidiagonals.

Examples

			The array starts as follows: (Elements in column 0 are also equal to the row index.)
col.0|  1 |  2 |  3  |  4  |  5  |  6 |  7 |  8  |  9  |  10 |  11 | 12 |  13 |  14
-----+----+----+-----+-----+-----+----+----+-----+-----+-----+-----+----+-----+-----
   0    0     0     0     0     0    0    0     0     0     0     0    0     0     0
   1   11    21    32    56  1130   50   50    50    50    50    50   50    50    50
   2   22    44   816  1548 18160  160   70    70    70    70    70   70    70    70
   3   33    69  1554 15100    70   70   70    70    70    70    70   70    70    70
   4   44   816  1548 18160   160   70   70    70    70    70    70   70    70    70
   5   55  1025    80    80    80   80   80    80    80    80    80   80    80    80
   6   66  1236  1236  1236  1236 1236 1236  1236  1236  1236  1236 1236  1236  1236
   7   77  1449 18144 18128 20128  130   40    40    40    40    40   40    40    40
   8   88  1664 17144 17112  1214   88 1664 17144 17112  1214    88 1664 17144 17112
   9   99  1881  1864 19192 22162 1348 1696 22324  1396 19162 19108  190   100    10
  10   10    10    10    10    10   10   10    10    10    10    10   10    10    10
  11   21    32    56  1130    50   50   50    50    50    50    50   50    50    50
  ...  ...  ...
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 -> ...
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.
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.
		

Crossrefs

Cf. A380873 (iterated function), A007953 (sum of digits), A007954 (product of digits).

Cf. A271220 (row 6), A271268 (row 8).

Programs

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))
A380872_array(rows=9, cols=rows)=Mat(vectorv(rows,i,A380872_row(i-1, cols)))

Formula

A(r,0) = r; A(r,n+1) = A380873(A(r,n)) = concat(A007953(A(r,n)), A007954(A(r,n))).