A397414 - OEIS

%I #9 Jun 26 2026 09:54:58

%S 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,

%T 49,52,56,17,19,23,85,87,93,95,111,64,77,82,86,88,91,94,98,104,29,31,

%U 37,115,125,135,145,153,155,159,38,41,43,47,76,112,118,121,122

%N 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).

%C 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.

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

%H Scott R. Shannon, <a href="/A397414/b397414.txt">Table of n, a(n) for n = 1..10000</a>

%H Scott R. Shannon, <a href="/A397414/a397414.png">Image of the first 50000 terms</a>. The green line is a(n) = n.

%e The triangle begins:

%e 2;

%e 3, 5;

%e 4, 8, 14;

%e 9, 15, 25, 27;

%e 7, 16, 22, 26, 28;

%e 45, 51, 57, 69, 75, 81;

%e 11, 13, 32, 44, 49, 52, 56;

%e 17, 19, 23, 85, 87, 93, 95, 111;

%e 64, 77, 82, 86, 88, 91, 94, 98, 104;

%e 29, 31, 37, 115, 125, 135, 145, 153, 155, 159;

%e 38, 41, 43, 47, 76, 112, 118, 121, 122, 128, 133;

%e 39, 53, 65, 117, 185, 195, 201, 207, 213, 219, 221, 225;

%e .

%e 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.

%Y Cf. A098550, A064413, A397219.

%K nonn,tabl,look

%O 1,1

%A _Scott R. Shannon_, Jun 24 2026