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Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.
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%I #5 Jun 17 2023 14:22:53

%S 1,4,4,9,1,9,2,17,17,2,3,7,1,7,3,4,5,8,8,5,4,5,4,9,1,9,4,5,8,2,3,11,

%T 11,3,2,8,16,5,11,6,1,6,11,5,16,1,17,7,4,9,9,4,7,17,1,1,4,17,10,6,1,6,

%U 10,17,4,1,1,4,9,14,5,15,15,5,14,9,4,1

%N Square array A(n, k), n, k > 0, read by antidiagonals; A(n, k) is the least e > 0 such that n^e and k^e have the same initial digit, or -1 if no such e exists.

%C Conjecture: all terms are positive.

%H Rémy Sigrist, <a href="/A363683/a363683.png">Colored representation of the table for n, k <= 1200</a>

%F A(n, k) = A(k, n).

%F A(10*n, k) = A(n, k).

%F A(n, n) = 1.

%F A(n, 1) = A098174(n).

%e Array A(n, k) begins:

%e n\k | 1 2 3 4 5 6 7 8 9 10 11 12

%e ----+-----------------------------------------------

%e 1 | 1 4 9 2 3 4 5 8 16 1 1 1

%e 2 | 4 1 17 7 5 4 2 5 17 4 4 9

%e 3 | 9 17 1 8 9 3 11 7 17 9 16 5

%e 4 | 2 7 8 1 11 6 4 10 14 2 2 2

%e 5 | 3 5 9 11 1 9 6 5 4 3 3 3

%e 6 | 4 4 3 6 9 1 15 7 11 4 4 10

%e 7 | 5 2 11 4 6 15 1 17 18 5 5 4

%e 8 | 8 5 7 10 5 7 17 1 18 8 28 6

%e 9 | 16 17 17 14 4 11 18 18 1 16 23 8

%e 10 | 1 4 9 2 3 4 5 8 16 1 1 1

%e 11 | 1 4 16 2 3 4 5 28 23 1 1 1

%e 12 | 1 9 5 2 3 10 4 6 8 1 1 1

%o (PARI) A(n,k) = { for (e = 1, oo, if (digits(n^e)[1]==digits(k^e)[1], return (e));); }

%Y Cf. A000030, A073601, A088995, A098174, A359698.

%K nonn,base,tabl

%O 1,2

%A _Rémy Sigrist_, Jun 15 2023