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A381965 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the multiplicative digital root of x (A031347).

%I A381965 #9 Mar 14 2025 21:20:24
%S A381965 0,1,2,3,4,5,6,7,8,9,10,0,11,1,12,2,13,3,14,4,15,5,16,6,17,7,18,8,19,
%T A381965 9,20,0,21,2,22,4,23,6,24,8,25,10,0,26,12,2,27,14,4,28,16,6,29,18,8,
%U A381965 30,0,31,3,32,6,33,9,34,12,2,35,15,5,36,18,8,37,21,2
%N A381965 Irregular triangle read by rows, where row n lists the iterates of f(x), starting at x = n until f(x) < 10, where f(x) is the multiplicative digital root of x (A031347).
%H A381965 Paolo Xausa, <a href="/A381965/b381965.txt">Table of n, a(n) for n = 0..11256</a> (rows 0..3500 of triangle, flattened).
%F A381965 T(n,0) = n and, for k = 1..A031346(n), T(n,k) = A031347(T(n,k-1)).
%e A381965 Triangle begins:
%e A381965   n\k|  0   1   2
%e A381965   ---------------
%e A381965    0 |  0;
%e A381965    1 |  1;
%e A381965    2 |  2;
%e A381965    3 |  3;
%e A381965    4 |  4;
%e A381965    5 |  5;
%e A381965    6 |  6;
%e A381965    7 |  7;
%e A381965    8 |  8;
%e A381965    9 |  9;
%e A381965   10 | 10,  0;
%e A381965   11 | 11,  1;
%e A381965   12 | 12,  2;
%e A381965   13 | 13,  3;
%e A381965   14 | 14,  4;
%e A381965   15 | 15,  5;
%e A381965   16 | 16,  6;
%e A381965   17 | 17,  7;
%e A381965   18 | 18,  8;
%e A381965   19 | 19,  9;
%e A381965   20 | 20,  0;
%e A381965   21 | 21,  2;
%e A381965   22 | 22,  4;
%e A381965   23 | 23,  6;
%e A381965   24 | 24,  8;
%e A381965   25 | 25, 10,  0;
%e A381965   ...
%t A381965 A381965row[n_] := NestWhileList[Times @@ IntegerDigits[#] &, n, # >= 10 &];
%t A381965 Array[A381965row, 50, 0]
%Y A381965 Cf. A031346 (row lengths - 1), A031347 (right border), A381966 (row sums).
%Y A381965 Cf. A381962, A381963.
%K A381965 nonn,tabf,base,easy
%O A381965 0,3
%A A381965 _Paolo Xausa_, Mar 11 2025