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3-almost prime triangle, read by rows.
2

%I #6 Mar 30 2012 18:40:41

%S 1,1,7,1,11,9,1,17,3,5,1,19,8,4,14,1,26,2,6,12,15,1,27,18,10,20,30,22,

%T 1,29,13,31,21,23,40,28,1,41,25,38,32,34,16,36,39,1,43,33,35,57,42,24,

%U 44,48,50

%N 3-almost prime triangle, read by rows.

%C This is to 3-almost primes (A014612) as A124883 is to semiprimes (A001358). 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 3-almost prime (A014612). Consider row 2. Starting with T(1,2) = 1, the least integer we can add to 1 and get a 3-almost prime is 7, since 1 + 8 = 8 = 2^3 is 3-almost prime. Consider row 3. Starting with T(1,3) = 1, the least integer we can add to 1 and get a 3-almost prime is 7, but we've already used that. The least unused integer that works is 11, since 1 + 11 = 12 = 2^2 * 3 is 3-almost prime. 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.

%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 A014612).

%e Triangle begins:

%e 1.

%e 1..7.

%e 1.11..9.

%e 1.17..3..5.

%e 1.19..8..4.14.

%e 1.26..2..6.12.15.

%e 1.27.18.10.20.30.22.

%e 1.29.13.31.21.23.40.28.

%e 1.41.25.38.32.34.16.36.39.

%e 1.43.33.35.57.42.24.44.48.50.

%Y Cf. A001358, A014612, A036440, A051237, A124883.

%K easy,nonn,tabl

%O 1,3

%A _Jonathan Vos Post_, Nov 12 2006