A343825 Table read by antidiagonals upward: T(n,k) is the least m such that there exists a sequence k = b_1 <= b_2 <= ... <= b_t = m such that no term appears n or more times and the product of the sequence is of the form c^n, where c is an integer; n >= 1 and k >= 0.
0, 0, 1, 0, 1, 2, 0, 1, 6, 3, 0, 1, 4, 8, 4, 0, 1, 4, 6, 4, 5, 0, 1, 4, 6, 9, 10, 6, 0, 1, 4, 6, 4, 10, 12, 7, 0, 1, 4, 6, 8, 10, 12, 14, 8, 0, 1, 4, 6, 4, 10, 9, 14, 15, 9, 0, 1, 4, 6, 8, 10, 9, 14, 8, 9, 10, 0, 1, 4, 6, 4, 10, 12, 14, 15, 16, 18, 11, 0, 1, 4
Offset: 1
Examples
Table begins:
n\k | 0 1 2 3 4 5 6 7 8 9 10
------+--------------------------------------
1 | 0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10
2 | 0, 1, 6, 8, 4, 10, 12, 14, 15, 9, 18
3 | 0, 1, 4, 6, 9, 10, 12, 14, 8, 16, 15
4 | 0, 1, 4, 6, 4, 10, 9, 14, 15, 9, 18
5 | 0, 1, 4, 6, 8, 10, 9, 14, 12, 15, 16
6 | 0, 1, 4, 6, 4, 10, 12, 14, 8, 9, 15
7 | 0, 1, 4, 6, 8, 10, 9, 14, 12, 15, 16
8 | 0, 1, 4, 6, 4, 10, 9, 14, 12, 9, 16
Specifically,
T(2,3) = 8 because 3 * 6 * 8 = 12^2,
T(3,3) = 6 because 3 * 4^2 * 6^2 = 12^3,
T(3,5) = 10 because 5 * 6 * 9 * 10^2 = 30^3,
T(4,6) = 9 because 6^2 * 8^2 * 9^3 = 36^4, and
T(4,9) = 9 because 9^2 = 3^4.
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