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A216653 Number A(n,k) of n-digit k-th powers; square array A(n,k), n>=1, k>=1, read by antidiagonals.

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