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A124883 Semiprime triangle, read by rows.

%I A124883 #20 Apr 08 2025 12:24:00
%S A124883 1,1,3,1,5,4,1,8,2,7,1,9,6,15,10,1,13,12,14,11,22,1,20,18,16,17,21,25,
%T A124883 1,24,27,19,30,28,23,26,1,32,33,29,36,38,31,34,35,1,37,40,42,43,39,46,
%U A124883 41,44,47,1,45,48,58,53,62,49,57,54,52,59
%N A124883 Semiprime triangle, read by rows.
%D A124883 R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.
%D A124883 M. J. Kenney, "Student Math Notes." NCTM News Bulletin. Nov. 1986.
%H A124883 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriangle.html">Prime Triangle</a>.
%F A124883 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 A124883 The n-th row is of length n. Each value is the smallest previously unused natural number such that the sum of every pair of adjacent values in the triangle is a semiprime (A001358).
%e A124883 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 A124883 Triangle begins:
%e A124883   1;
%e A124883   1,  3;
%e A124883   1,  5,  4;
%e A124883   1,  8,  2,  7;
%e A124883   1,  9,  6, 15, 10;
%e A124883   1, 13, 12, 14, 11, 22;
%e A124883   1, 20, 18, 16, 17, 21, 25;
%e A124883   1, 24, 27, 19, 30, 28, 23, 26;
%e A124883   1, 32, 33, 29, 36, 38, 31, 34, 35;
%e A124883   1, 37, 40, 42, 43, 39, 46, 41, 44, 47;
%e A124883   1, 45, 48, 58, 53, 62, 49, 57, 54, 52, 59;
%e A124883   ...
%Y A124883 Cf. A001358, A036440, A051237.
%K A124883 easy,nonn,tabl
%O A124883 1,3
%A A124883 _Jonathan Vos Post_, Nov 11 2006
%E A124883 Terms corrected by _Alois P. Heinz_, Apr 08 2025