A343881 Table read by antidiagonals upward: T(n,k) is the least integer m > k such that k^x * m^y = c^n for some positive integers c, x, and y where x < n and y < n; n >= 2, k >= 1.

4, 8, 8, 4, 4, 12, 32, 4, 9, 9, 4, 4, 9, 16, 20, 128, 4, 9, 8, 25, 24, 4, 4, 9, 8, 20, 36, 28, 8, 4, 9, 8, 25, 24, 49, 18, 4, 4, 9, 8, 20, 36, 28, 27, 16, 2048, 4, 9, 8, 25, 24, 49, 18, 24, 40, 4, 4, 9, 8, 20, 36, 28, 16, 12, 80, 44, 8192, 4, 9, 8, 25, 24, 49

Offset: 2

Author

Peter Kagey, May 02 2021

Keywords

• nonn
• tabl

Comments

For prime p, the p-th row consists of distinct integers.

Conjecture: T(p,k) = A064549(k) for fixed k > 1 and sufficiently large p.

Examples

			Table begins:
  n\k|    1  2   3   4   5   6   7   8   9   10
-----+-----------------------------------------
   2 |    4, 8, 12,  9, 20, 24, 28, 18, 16,  40
   3 |    8, 4,  9, 16, 25, 36, 49, 27, 24,  80
   4 |    4, 4,  9,  8, 20, 24, 28, 18, 12,  40
   5 |   32, 4,  9,  8, 25, 36, 49, 16, 27, 100
   6 |    4, 4,  9,  8, 20, 24, 28,  9, 16,  40
   7 |  128, 4,  9,  8, 25, 36, 49, 16, 27, 100
   8 |    4, 4,  9,  8, 20, 24, 28, 16, 12,  40
   9 |    8, 4,  9,  8, 25, 36, 49, 16, 24,  80
  10 |    4, 4,  9,  8, 20, 24, 28, 16, 16,  40
  11 | 2048, 4,  9,  8, 25, 36, 49, 16, 27, 100
T(2, 3) = 12 with  3   * 12   =  6^2.
T(3,10) = 80 with 10^2 * 80   = 20^3.
T(4, 5) = 20 with  5^2 * 20^2 = 10^4.
T(5, 1) = 32 with  1   * 32   =  2^5.
T(6, 8) =  9 with  8^2 *  9^3 =  6^6.
		

Crossrefs

Rows: A072905 (n=2), A277781 (n=3).

Cf. A064549, A343825.

Formula

T(n,1) = 2^A020639(n).