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Irregular triangle read by rows in which row n lists the numbers that divide the sum of the digits of their n-th powers.
1

%I #29 Jul 15 2022 21:48:56

%S 1,1,2,3,4,5,6,7,8,9,1,2,3,9,1,2,3,8,9,17,18,26,27,1,3,6,7,9,22,25,28,

%T 36,1,3,9,28,35,36,46,1,2,3,7,9,18,23,45,54,64,1,3,6,9,12,15,18,27,31,

%U 34,43,53,58,68,1,3,5,6,9,15,27,46,54,63

%N Irregular triangle read by rows in which row n lists the numbers that divide the sum of the digits of their n-th powers.

%C For the proof of finiteness of rows, see comments in A309017.

%C It appears that the right column is A046000.

%e Triangle begins:

%e n=0: 1;

%e n=1: 1, 2, 3, 4, 5, 6, 7, 8, 9;

%e n=2: 1, 2, 3, 9;

%e n=3: 1, 2, 3, 8, 9, 17, 18, 26, 27;

%e n=4: 1, 3, 6, 7, 9, 22, 25, 28, 36;

%e n=5: 1, 3, 9, 28, 35, 36, 46;

%e n=6: 1, 2, 3, 7, 9, 18, 23, 45, 54, 64;

%e n=7: 1, 3, 6, 9, 12, 15, 18, 27, 31, 34, 43, 53, 58, 68;

%e n=8: 1, 3, 5, 6, 9, 15, 27, 46, 54, 63;

%e n=9: 1, 2, 3, 6, 7, 9, 16, 27, 36, 54, 71, 81;

%e n=10: 1, 3, 5, 6, 9, 18, 36, 82, 85, 94, 97, 106, 117;

%e ...

%o (Python)

%o def ok(k, n): return sum(map(int, str(k**n)))%k==0

%o def row(n):

%o d, lim = 1, 1

%o while lim < n*9*d: d, lim = d+1, lim*10

%o yield from [k for k in range(1, lim+1) if ok(k, n)]

%o print([an for n in range(9) for an in row(n)]) # _Michael S. Branicky_, Jul 06 2022

%Y Cf. A046000, A152147, A309017.

%Y Row lengths are A355563.

%K tabf,nonn,base

%O 0,3

%A _Mohammed Yaseen_, Jun 30 2022