A363683 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.

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

Offset: 1

Author

Rémy Sigrist, Jun 15 2023

Keywords

• nonn
• base
• tabl

Comments

Conjecture: all terms are positive.

Examples

			Array A(n, k) begins:
  n\k |  1   2   3   4   5   6   7   8   9  10  11  12
  ----+-----------------------------------------------
    1 |  1   4   9   2   3   4   5   8  16   1   1   1
    2 |  4   1  17   7   5   4   2   5  17   4   4   9
    3 |  9  17   1   8   9   3  11   7  17   9  16   5
    4 |  2   7   8   1  11   6   4  10  14   2   2   2
    5 |  3   5   9  11   1   9   6   5   4   3   3   3
    6 |  4   4   3   6   9   1  15   7  11   4   4  10
    7 |  5   2  11   4   6  15   1  17  18   5   5   4
    8 |  8   5   7  10   5   7  17   1  18   8  28   6
    9 | 16  17  17  14   4  11  18  18   1  16  23   8
   10 |  1   4   9   2   3   4   5   8  16   1   1   1
   11 |  1   4  16   2   3   4   5  28  23   1   1   1
   12 |  1   9   5   2   3  10   4   6   8   1   1   1
		

Links

• Rémy Sigrist, Colored representation of the table for n, k <= 1200

Crossrefs

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

Programs

PARI

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

Formula

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

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

A(n, n) = 1.

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