A397414 Triangle read by rows: T(1,1) = 2; for n > 1, 1 <= k <= n, T(n,k) is the smallest unused positive number that has a common factor with at least one term in row T(n-2) but has no common factor with any term in row T(n-1).

2, 3, 5, 4, 8, 14, 9, 15, 25, 27, 7, 16, 22, 26, 28, 45, 51, 57, 69, 75, 81, 11, 13, 32, 44, 49, 52, 56, 17, 19, 23, 85, 87, 93, 95, 111, 64, 77, 82, 86, 88, 91, 94, 98, 104, 29, 31, 37, 115, 125, 135, 145, 153, 155, 159, 38, 41, 43, 47, 76, 112, 118, 121, 122

Offset

1,1

Comments

The sequence is a version of the Yellowstone permutation A098550 where the number of terms that must satisfy the common factor criteria increases by one for each group/row of terms. However, unlike that sequence, it appears that many numbers are never terms - after 50000 terms the numbers 6, 10, 12, 18, 20, 21, 24, 30 have not appeared.

For the terms studied the primes appear in their natural order.

Links

Scott R. Shannon, Table of n, a(n) for n = 1..10000

Scott R. Shannon, Image of the first 50000 terms. The green line is a(n) = n.

Example

The triangle begins:
2;
3, 5;
4, 8, 14;
9, 15, 25, 27;
7, 16, 22, 26, 28;
45, 51, 57, 69, 75, 81;
11, 13, 32, 44, 49, 52, 56;
17, 19, 23, 85, 87, 93, 95, 111;
64, 77, 82, 86, 88, 91, 94, 98, 104;
29, 31, 37, 115, 125, 135, 145, 153, 155, 159;
38, 41, 43, 47, 76, 112, 118, 121, 122, 128, 133;
39, 53, 65, 117, 185, 195, 201, 207, 213, 219, 221, 225;
.
T(5,1..5) = a(11..15) = 7, 16, 22, 26, 28 as these are the smallest unused numbers that all share a common factor with at least one term in T(3) = 4, 8, 14, while sharing no common factor with any term in T(4) = 9, 15, 25, 27.

Crossrefs

Cf. A098550, A064413, A397219.

Sequence in context: A245608 A244154 A182395 * A244983 A059450 A255972

Adjacent sequences: A397407 A397408 A397411 * A397416 A397417 A397418

Keyword

nonn,tabl,look

Author

Scott R. Shannon, Jun 24 2026

Status

approved