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A124883
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Semiprime triangle, read by rows.
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1
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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, 1, 24, 25, 26, 23, 28, 27, 30, 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, 61, 50
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OFFSET
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1,3
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REFERENCES
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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.
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LINKS
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FORMULA
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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).
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EXAMPLE
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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).
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.
1.13.12.21.11.22.
1.14.19.16.17.18.20.
1.24.25.26.23.28.27.30.
1.32.33.29.36.38.31.34.35.
1.37.40.42.43.39.46.41.44.47.
1.45.48.58.53.62.49.57.54.61.50
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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