A362034 Triangle read by rows: T(n,0) = T(n,n) = 2, 0 < k < n: T(n,k) = smallest prime not less than T(n-1,k) + T(n-1,k-1).

2, 2, 2, 2, 5, 2, 2, 7, 7, 2, 2, 11, 17, 11, 2, 2, 13, 29, 29, 13, 2, 2, 17, 43, 59, 43, 17, 2, 2, 19, 61, 103, 103, 61, 19, 2, 2, 23, 83, 167, 211, 167, 83, 23, 2, 2, 29, 107, 251, 379, 379, 251, 107, 29, 2, 2, 31, 137, 359, 631, 761, 631, 359, 137, 31, 2

Offset: 0

Author

Jack Braxton, Apr 05 2023

Keywords

• nonn
• tabl

Comments

In order to get the next number in the row, you add the two numbers above it, and find the next prime.

3 is the only prime number that never shows up.

5 is the only prime number that only shows up once; every prime number above 5 shows up at least twice.

Examples

			Triangle begins:
      k=0  1   2   3   4   5   6   7   8  9 10
  n=0:  2
  n=1:  2  2
  n=2:  2  5   2
  n=3:  2  7   7   2
  n=4:  2 11  17  11   2
  n=5:  2 13  29  29  13   2
  n=6:  2 17  43  59  43  17   2
  n=7:  2 19  61 103 103  61  19   2
  n=8:  2 23  83 167 211 167  83  23   2
  n=9:  2 29 107 251 379 379 251 107  29  2
 n=10:  2 31 137 359 631 761 631 359 137 31  2
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 0..11475 (rows 0..150, flattened)

Crossrefs

Cf. A000040, A007318, A132403, A199333.

Programs

Maple

for n from 0 to 10 do
  T[n,0]:= 2: T[n,n]:= 2:
  for k from 1 to n-1 do
    T[n,k]:= nextprime(T[n-1,k-1]+T[n-1,k]-1)
  od
od:
for n from 0 to 10 do
  seq(T[n,k],k=0..n)
od; # Robert Israel, Apr 05 2023

Mathematica

T[n_, 0] := T[n, n] = 2; T[n_, k_] := T[n, k] = NextPrime[T[n - 1, k - 1] + T[n - 1, k] - 1]; Table[T[n, k], {n, 0, 10}, {k, 0, n}] // Flatten (* Michael De Vlieger, Apr 06 2023, after Maple *)

PARI

T(n,k) = if (n==0, 2, if (k==0, 2, if (k==n, 2, nextprime(T(n-1,k-1) + T(n-1,k))))); \\ Michel Marcus, Apr 07 2023

Formula

T(n,k) = A007918(T(n-1,k-1) + T(n-1,k)) for 0 < k < n. - Robert Israel, Apr 05 2023