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%I #7 Dec 05 2013 19:55:46
%S 1,2,8,3,4,18,5,6,7,9261000,9,10,11,12,329422500,13,14,15,16,17,
%T 13456677968449745006250,19,20,21,22,23,24,
%U 11022732501667945875061568782593750,25,26,27,28,29,30,31
%N 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.
%C Another rearrangement of the natural numbers in which the product of next n numbers is an n-th power.
%C 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
%e Triangle begins:
%e 1
%e 2 8
%e 3 4 18
%e 5 6 7 9261000
%e 9 10 11 12 329422500
%e ...
%Y Cf. A076027, A076028, A076029, A076030, A076031, A076095, A076096, A076097, A076098.
%K nonn,tabl
%O 0,2
%A _Amarnath Murthy_, Oct 08 2002
%E Corrected and extended by _Sam Alexander_, Dec 31 2003
%E Edited by _Ray Chandler_, May 09 2007
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