OFFSET
1,1
COMMENTS
The sequence is a version of the Enots Wolley sequence A336957 where the number of terms that must satisfy the common factor criteria increases by one for each group/row of terms. Unlike that sequence terms can exist that are primes or prime powers as only one number in each row is required to have a factor that is not a factor of any number in the previous row. For the terms studied this appears to happen when a simple greedy algorithm is used to find the terms, and it is likely this is true for all rows.
Curiously all the multiples of 10 do not appear to be terms. This can be seen by noting the rows with numbers that contain 2 and 5 as factors - there are two rows that contain 2 followed by two rows that contain 5, and given that pattern repeats indefinitely, any multiple of 10 can never be a term.
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 100000 terms. The green line is a(n) = n.
EXAMPLE
The triangle begins:
2;
4, 6;
3, 9, 15;
5, 25, 35, 55;
7, 11, 14, 22, 28;
8, 12, 16, 18, 24, 26;
13, 27, 39, 45, 51, 57, 65;
17, 19, 85, 95, 115, 119, 125, 133;
23, 46, 49, 56, 77, 92, 98, 112, 154;
32, 33, 36, 44, 48, 52, 54, 58, 62, 64;
29, 31, 75, 81, 87, 93, 111, 117, 123, 129, 135;
37, 41, 43, 175, 185, 205, 215, 235, 245, 259, 265, 287;
...
T(5,1..5) = a(11..15) = 7, 11, 14, 22, 28 as these are the smallest unused numbers that all share a common factor with at least one term in T(4) = 5, 25, 35, 55, while sharing no common factor with any term in T(3) = 3, 9, 15.
CROSSREFS
AUTHOR
Scott R. Shannon, Jun 26 2026
STATUS
approved