A124883 Semiprime triangle, read by rows.

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, 1, 24, 27, 19, 30, 28, 23, 26, 1, 32, 33, 29, 36, 38, 31, 34, 35, 1, 37, 40, 42, 43, 39, 46, 41, 44, 47, 1, 45, 48, 58, 53, 62, 49, 57, 54, 52, 59

Offset: 1

Author

Jonathan Vos Post, Nov 11 2006

Keywords

• easy
• nonn
• tabl

Examples

			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).
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.
Triangle begins:
  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;
  1, 24, 27, 19, 30, 28, 23, 26;
  1, 32, 33, 29, 36, 38, 31, 34, 35;
  1, 37, 40, 42, 43, 39, 46, 41, 44, 47;
  1, 45, 48, 58, 53, 62, 49, 57, 54, 52, 59;
  ...
		

References

• R. K. Guy, Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, p. 106, 1994.
• M. J. Kenney, "Student Math Notes." NCTM News Bulletin. Nov. 1986.

Links

• Eric Weisstein's World of Mathematics, Prime Triangle.

Crossrefs

Cf. A001358, A036440, A051237.

Formula

T(n,1) = 1 for all natural numbers n. For n>1 and 1T(n,i) for iT(r,s) for rA001358}.

Extensions

Terms corrected by Alois P. Heinz, Apr 08 2025