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A124883 Semiprime triangle, read by rows. 1
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 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,3

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

Table of n, a(n) for n=1..66.

Eric Weisstein's World of Mathematics, Prime Triangle.

FORMULA

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).

EXAMPLE

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

CROSSREFS

Cf. A001358, A036440, A051237.

Sequence in context: A104734 A029655 A110813 * A131809 A016574 A210560

Adjacent sequences:  A124880 A124881 A124882 * A124884 A124885 A124886

KEYWORD

easy,nonn,tabl

AUTHOR

Jonathan Vos Post, Nov 11 2006

STATUS

approved

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Last modified April 20 13:52 EDT 2014. Contains 240806 sequences.