A076099 Triangle read by rows in which the n-th row contains n numbers not occurring in the previous rows whose product is an n-th power. The first (n-1) numbers of the n-th row are the smallest number not occurring earlier and the n-th term is chosen to satisfy the requirement.

1, 2, 8, 3, 4, 18, 5, 6, 7, 9261000, 9, 10, 11, 12, 329422500, 13, 14, 15, 16, 17, 13456677968449745006250, 19, 20, 21, 22, 23, 24, 11022732501667945875061568782593750, 25, 26, 27, 28, 29, 30, 31

Offset: 1

Author

Amarnath Murthy, Oct 08 2002

Keywords

• nonn
• tabl

Comments

Another rearrangement of the natural numbers in which the product of next n numbers is an n-th power.

The first n-1 elements of the n-th group are the smallest n-1 numbers that haven't already appeared, say u1, u2, ..., u_(n-1) and let u_n be the unknown final element of the n-th group. Let u1*u2*u3*...*u_(n-1) = (p1^e1)(p2^e2)...(pr^er). Then u_n = product(i=1 to r) p_i^(ei + n*floor(ei/n) - n) ...unless this has already appeared in the sequence (probably this never happens). More simply, I conjecture that u_n = product(i=1 to r) p_i^(ei - n). - Sam Alexander, Dec 31 2003

Examples

			Triangle begins:
1
2 8
3 4 18
5 6 7 9261000
9 10 11 12 329422500
...
		

Crossrefs

Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076096, A076097, A076098.

Extensions

Corrected and extended by Sam Alexander, Dec 31 2003

Edited by Ray Chandler, May 09 2007

Offset corrected by Sean A. Irvine, Mar 19 2025