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A124886 3-almost prime triangle, read by rows.

%I A124886 #11 Apr 08 2025 03:32:58
%S A124886 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 A124886 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 A124886 44,48,50
%N A124886 3-almost prime triangle, read by rows.
%C A124886 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 A124886 R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.
%D A124886 M. J. Kenney, "Student Math Notes." NCTM News Bulletin. Nov. 1986.
%H A124886 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PrimeTriangle.html">Prime Triangle</a>.
%F A124886 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 A124886 Triangle begins:
%e A124886   1
%e A124886   1  7
%e A124886   1 11  9
%e A124886   1 17  3  5
%e A124886   1 19  8  4 14
%e A124886   1 26  2  6 12 15
%e A124886   1 27 18 10 20 30 22
%e A124886   1 29 13 31 21 23 40 28
%e A124886   1 41 25 38 32 34 16 36 39
%e A124886   1 43 33 35 57 42 24 44 48 50
%Y A124886 Cf. A001358, A014612, A036440, A051237, A124883.
%K A124886 easy,nonn,tabl
%O A124886 1,3
%A A124886 _Jonathan Vos Post_, Nov 12 2006