A397219 Pascal-like triangle read by rows: top entry is 2; thereafter fill in entries left to right, top to bottom, with smallest number m that is not yet in the sequence and has a common factor with any earlier entry that is adjacent to it in the triangle.

2, 4, 6, 8, 10, 12, 14, 16, 18, 3, 7, 28, 20, 15, 9, 21, 35, 30, 5, 45, 24, 27, 42, 40, 25, 50, 36, 22, 33, 39, 26, 60, 55, 44, 32, 34, 11, 66, 48, 38, 70, 77, 56, 46, 52, 88, 99, 51, 54, 58, 84, 49, 98, 62, 64, 68, 72, 57, 63, 78, 74, 112, 91, 104, 76, 80, 17, 102

Offset

1,1

Comments

The formal definition is as follows:

Denote the triangle by [T(n,k), n >= 0, 0 <= k <= n]. Then T(0,0)=2;

for n>0, T(n,0) = smallest unused number m such that GCD(m,T(n-1,0)) > 1;

for n>0 and 1<=k<=n-1, T(n,k) = smallest unused number m such that GCD(m,T(n-1,k-1)) > 1, GCD(m,T(n-1,k))>1, and GCD(m, T(n,k-1))>1; and

for n>0, T(n,n) = smallest unused number m such that GCD(m,T(n-1,n-1)) > 1 and GCD(m, T(n,n-1))>1.

This is a kind of hybrid of the EKG sequence and Pascal's triangle.

It is conjectured that every number greater than 1 appears (cf. A397218).

Links

Michael S. Branicky, Table of n, a(n) for n = 1..20000

Michael S. Branicky, Table of n, a(n) for n = 1..10^6

Scott R. Shannon, Image of the first 1000 terms on the triangle. The number colors are graduated across the spectrum, from red to violet, to show their relative size.

Scott R. Shannon, Image of the first 100000 terms on the triangle.

Example

Triangle begins:
                       2,
                     4, 6,
                   8, 10, 12,
                 14, 16, 18, 3,
                7, 28, 20, 15, 9,
              21, 35, 30, 5, 45, 24,
            27, 42, 40, 25, 50, 36, 22,
          33, 39, 26, 60, 55, 44, 32, 34,
        11, 66, 48, 38, 70, 77, 56, 46, 52,
...

Mathematica

Block[{c, k, t, u, rows}, rows = 12; c[_] := False; t[_, _] := 1; t[0, 0] = 2; c[2] = True; u = 3; {t[0, 0]}~Join~Reap[Do[k = u; Which[j == 0, While[Or[c[k], CoprimeQ[t[i - 1, 0], k] ], k++], j == i, While[Or[c[k], AnyTrue[{t[i, j - 1], t[i - 1, j - 1]}, CoprimeQ[#, k] &] ], k++], True, While[Or[c[k], AnyTrue[{t[i, j - 1], t[i - 1, j], t[i - 1, j - 1]}, CoprimeQ[#, k] &]], k++]]; Sow[Set[t[i, j], k] ]; c[k] = True; If[k == u, While[c[u], u++] ], {i, rows}, {j, 0, i}] ][[-1, 1]] ] (* Michael De Vlieger, Jun 19 2026 *)

Prog

(Python)
from math import gcd
from itertools import count, islice
def A397219_gen():
    T, m, am, aset = {(0, 0): 2}, 3, 2, set()
    for n in count(1):
        for k in range(n+1):
            yield am
            aset.add(am)
            if k == 0: neighs = {T[n-1, 0]}
            elif 1 <= k < n: neighs = {T[n-1, k-1], T[n-1, k], T[n, k-1]}
            else: neighs = {T[n-1, k-1], T[n, k-1]}
            am = next(k for k in count(m) if k not in aset and all(gcd(v, k)>1 for v in neighs))
            T[n, k] = am
print(list(islice(A397219_gen(), 68))) # Michael S. Branicky, Jun 19 2026

Crossrefs

Cf. A064413 (EKG), A007318 (Pascal), A397218.

For leading diagonal see A397169.

Sequence in context: A169805 A340479 A263042 * A230099 A098727 A182324

Adjacent sequences: A397212 A397217 A397218 * A397221 A397223 A397224

Keyword

nonn,tabl,new

Author

Scott R. Shannon, Jun 19 2026

Extensions

Edited by N. J. A. Sloane, Jun 19 2026

Status

approved