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A124886
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3-almost prime triangle, read by rows.
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2
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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, 1, 29, 13, 31, 21, 23, 40, 28, 1, 41, 25, 38, 32, 34, 16, 36, 39, 1, 43, 33, 35, 57, 42, 24, 44, 48, 50
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OFFSET
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1,3
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COMMENTS
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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.
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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 A014612).
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EXAMPLE
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Triangle begins:
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.
1.29.13.31.21.23.40.28.
1.41.25.38.32.34.16.36.39.
1.43.33.35.57.42.24.44.48.50.
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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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