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%I #12 Dec 29 2018 07:51:22
%S 10,4,90,3,6,900,2,2,22,9000,2,2,5,68,90000,2,1,2,12,217,900000,2,1,1,
%T 4,25,683,9000000,2,0,1,3,8,53,2163,90000000,2,0,1,1,3,14,116,6837,
%U 900000000,2,0,1,1,2,6,25,249,21623,9000000000
%N Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.
%H Alois P. Heinz, <a href="/A216653/b216653.txt">Antidiagonals n = 1..141, flattened</a>
%e A(1,1) = 10: 0, 1, 2, 3, 4, 5, 6, 7, 8, 9.
%e A(1,2) = 4: 0, 1, 4, 9.
%e A(2,2) = 6: 16, 25, 36, 49, 64, 81.
%e A(3,3) = 5: 125, 216, 343, 512, 729.
%e A(4,4) = 4: 1296, 2401, 4096, 6561.
%e A(5,5) = 3: 16807, 32768, 59049.
%e A(6,6) = 3: 117649, 262144, 531441.
%e Square array A(n,k) begins:
%e :n\k| 1: 2: 3: 4: 5: 6: 7: 8
%e +---+--------------------------------------------
%e : 1 | 10, 4, 3, 2, 2, 2, 2, 2
%e : 2 | 90, 6, 2, 2, 1, 1, 0, 0
%e : 3 | 900, 22, 5, 2, 1, 1, 1, 1
%e : 4 | 9000, 68, 12, 4, 3, 1, 1, 1
%e : 5 | 90000, 217, 25, 8, 3, 2, 2, 1
%e : 6 | 900000, 683, 53, 14, 6, 3, 2, 1
%e : 7 | 9000000, 2163, 116, 25, 10, 5, 2, 2
%e : 8 | 90000000, 6837, 249, 43, 14, 7, 4, 2
%p r:= proc(n, k) local b; b:= iroot(n, k); b+`if`(b^k<n, 1, 0) end:
%p A:= (n, k)-> r(10^n, k) -r(10^(n-1), k) +`if`(n=1, 1, 0):
%p seq(seq(A(n, 1+d-n), n=1..d), d=1..10);
%Y Columns k=1-10 give: A063945, A062940, A062941, A102831, A216655, A216656, A216657, A216658, A216659, A216654.
%Y Main diagonal gives: A102690.
%K nonn,tabl,base
%O 1,1
%A _Alois P. Heinz_, Sep 12 2012
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