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

1,6

Comments

For each prime p, the p-th row is a permutation of the nonprime integers.

T(n,k) <= A343881(n,k).

Conjecture: T(p,k) = A071537(k) for fixed k and sufficiently large prime p.

Links

Table of n, a(n) for n=1..81.

Example

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.

Crossrefs

Row n: A001477 (n=1), A006255 (n=2), A277494 (n=3), A328045 (n=4).

Cf. A071537.

Sequence in context: A175197 A376983 A367795 * A339031 A367270 A365770

Adjacent sequences: A343822 A343823 A343824 * A343826 A343827 A343828

Keyword

nonn,tabl

Author

Peter Kagey, Apr 30 2021

Status

approved