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A355370 Irregular triangle read by rows in which row n lists the numbers that divide the sum of the digits of their n-th powers.

%I A355370 #29 Jul 15 2022 21:48:56
%S A355370 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 A355370 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 A355370 34,43,53,58,68,1,3,5,6,9,15,27,46,54,63
%N A355370 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 A355370 For the proof of finiteness of rows, see comments in A309017.
%C A355370 It appears that the right column is A046000.
%e A355370 Triangle begins:
%e A355370   n=0:  1;
%e A355370   n=1:  1, 2, 3,  4,  5,  6,  7,  8,  9;
%e A355370   n=2:  1, 2, 3,  9;
%e A355370   n=3:  1, 2, 3,  8,  9, 17, 18, 26, 27;
%e A355370   n=4:  1, 3, 6,  7,  9, 22, 25, 28, 36;
%e A355370   n=5:  1, 3, 9, 28, 35, 36, 46;
%e A355370   n=6:  1, 2, 3,  7,  9, 18, 23, 45, 54, 64;
%e A355370   n=7:  1, 3, 6,  9, 12, 15, 18, 27, 31, 34, 43,  53,  58, 68;
%e A355370   n=8:  1, 3, 5,  6,  9, 15, 27, 46, 54, 63;
%e A355370   n=9:  1, 2, 3,  6,  7,  9, 16, 27, 36, 54, 71,  81;
%e A355370   n=10: 1, 3, 5,  6,  9, 18, 36, 82, 85, 94, 97, 106, 117;
%e A355370   ...
%o A355370 (Python)
%o A355370 def ok(k, n): return sum(map(int, str(k**n)))%k==0
%o A355370 def row(n):
%o A355370     d, lim = 1, 1
%o A355370     while lim < n*9*d: d, lim = d+1, lim*10
%o A355370     yield from [k for k in range(1, lim+1) if ok(k, n)]
%o A355370 print([an for n in range(9) for an in row(n)]) # _Michael S. Branicky_, Jul 06 2022
%Y A355370 Cf. A046000, A152147, A309017.
%Y A355370 Row lengths are A355563.
%K A355370 tabf,nonn,base
%O A355370 0,3
%A A355370 _Mohammed Yaseen_, Jun 30 2022