A124886 - OEIS

A124886

3-almost prime triangle, read by rows.

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

1,3

COMMENTS

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.

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..55.

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 A014612).

EXAMPLE

Triangle begins:

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.

CROSSREFS