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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
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
1,2
COMMENTS
Conjecture: all terms are positive.
FORMULA
A(n, k) = A(k, n).
A(10*n, k) = A(n, k).
A(n, n) = 1.
A(n, 1) = A098174(n).
EXAMPLE
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
PROG
(PARI) A(n, k) = { for (e = 1, oo, if (digits(n^e)[1]==digits(k^e)[1], return (e)); ); }
CROSSREFS
KEYWORD
nonn,base,tabl
AUTHOR
Rémy Sigrist, Jun 15 2023
STATUS
approved