A101543 Triangle read by rows: First row = 2; n-th row (n>1) has n smallest positive integers not yet in the sequence such that each integer has a prime divisor in common with at least one element of the (n-1)st row.

2, 4, 6, 3, 8, 9, 10, 12, 14, 15, 5, 7, 16, 18, 20, 21, 22, 24, 25, 26, 27, 11, 13, 28, 30, 32, 33, 34, 17, 35, 36, 38, 39, 40, 42, 44, 19, 45, 46, 48, 49, 50, 51, 52, 54, 23, 55, 56, 57, 58, 60, 62, 63, 64, 65, 29, 31, 66, 68, 69, 70, 72, 74, 75, 76, 77, 37, 78, 80, 81, 82, 84

Offset

1,1

Comments

Is this a permutation of the integers >= 2?

It appears that for n > 2, row n+1 always begins with the primes p such that 2p appears in row n and the rest of row n+1 consists of the smallest composite numbers not already used. The only way this pattern can break down is if we have to skip a composite number because it doesn't share a factor with any number in the previous row. Let f(n) be the last number in row n. To prove that this pattern continues, it suffices to show that f(n) < (f(n-1)-f(n-2)+1)^2, because the prime factors of row n-1 include all primes <= f(n-1)-f(n-2) and any composite number x has a prime factor <= sqrt(x). I have checked that f(n) < (f(n-1)-f(n-2)+1)^2 for all n up to 10000. In fact for 1000 < n <= 10000, f(n) < (f(n-1)-f(n-2)-300)^2. - David Wasserman, Mar 27 2008

Links

Table of n, a(n) for n=1..72.

Example

7 is in the 5th row because it does not occur earlier and 14 is in the 4th row.

Crossrefs

Sequence in context: A251622 A073899 A232846 * A073900 A361629 A350033

Adjacent sequences: A101540 A101541 A101542 * A101544 A101545 A101546

Keyword

nonn,tabl

Author

Leroy Quet, Jan 25 2005

Extensions

More terms from David Wasserman, Mar 27 2008

Edited by N. J. A. Sloane, Apr 16 2008

Status

approved