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%I #7 Mar 30 2012 18:40:41
%S 1,1,3,1,5,4,1,8,2,7,1,9,6,15,10,1,13,12,21,11,22,1,14,19,16,17,18,20,
%T 1,24,25,26,23,28,27,30,1,32,33,29,36,38,31,34,35,1,37,40,42,43,39,46,
%U 41,44,47,1,45,48,58,53,62,49,57,54,61,50
%N Semiprime triangle, read by rows.
%D R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.
%D M. J. Kenney, "Student Math Notes." NCTM News Bulletin. Nov. 1986.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PrimeTriangle.html">Prime Triangle</a>.
%F T(n,1) = 1 for all natural numbers n. For n>1 and 1<k<n we have T(n,k) = min{j such that j<>T(n,i) for i<k and j<>T(r,s) for r<n and for all i<j we have T(i,j) + T(i,j-1) is in A001358).
%e The n-th row is of length n. Each value is the smallest previously unused natural number such that every pair of adjacent values in the triangle is semiprime (A001358).
%e Consider row 2. Starting with T(1,2) = 1, the least integer we can add to 1 and get a semiprime is 3, since 1 + 3 = 4 = 2^2 is semiprime. Consider row 3. Starting with T(1,3) = 1, the least integer we can add to 1 and get a semiprime is 1, but we've already used that. The next is 3, but we've used that. The least unused integer that works is 5, since 1 + 5 = 6 = 2 * 3 is semiprime. If we cross out ones from the triangle read by rows, what remains is a permutation of the natural number greater than 1. That is, every nonnegative integer appears in the triangle. The second column T(n,2) is monotone increasing.
%e Triangle begins:
%e 1.
%e 1..3.
%e 1..5..4.
%e 1..8..2..7.
%e 1..9..6.15.10.
%e 1.13.12.21.11.22.
%e 1.14.19.16.17.18.20.
%e 1.24.25.26.23.28.27.30.
%e 1.32.33.29.36.38.31.34.35.
%e 1.37.40.42.43.39.46.41.44.47.
%e 1.45.48.58.53.62.49.57.54.61.50
%Y Cf. A001358, A036440, A051237.
%K easy,nonn,tabl
%O 1,3
%A _Jonathan Vos Post_, Nov 11 2006
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