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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).

%I A362034 #46 Apr 15 2025 18:29:40
%S A362034 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,
%T A362034 2,19,61,103,103,61,19,2,2,23,83,167,211,167,83,23,2,2,29,107,251,379,
%U A362034 379,251,107,29,2,2,31,137,359,631,761,631,359,137,31,2
%N 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).
%C A362034 In order to get the next number in the row, you add the two numbers above it, and find the next prime.
%C A362034 3 is the only prime number that never shows up.
%C A362034 5 is the only prime number that only shows up once; every prime number above 5 shows up at least twice.
%H A362034 Michael De Vlieger, <a href="/A362034/b362034.txt">Table of n, a(n) for n = 0..11475</a> (rows 0..150, flattened)
%F A362034 T(n,k) = A007918(T(n-1,k-1) + T(n-1,k)) for 0 < k < n. - _Robert Israel_, Apr 05 2023
%e A362034 Triangle begins:
%e A362034       k=0  1   2   3   4   5   6   7   8  9 10
%e A362034   n=0:  2
%e A362034   n=1:  2  2
%e A362034   n=2:  2  5   2
%e A362034   n=3:  2  7   7   2
%e A362034   n=4:  2 11  17  11   2
%e A362034   n=5:  2 13  29  29  13   2
%e A362034   n=6:  2 17  43  59  43  17   2
%e A362034   n=7:  2 19  61 103 103  61  19   2
%e A362034   n=8:  2 23  83 167 211 167  83  23   2
%e A362034   n=9:  2 29 107 251 379 379 251 107  29  2
%e A362034  n=10:  2 31 137 359 631 761 631 359 137 31  2
%p A362034 for n from 0 to 10 do
%p A362034   T[n,0]:= 2: T[n,n]:= 2:
%p A362034   for k from 1 to n-1 do
%p A362034     T[n,k]:= nextprime(T[n-1,k-1]+T[n-1,k]-1)
%p A362034   od
%p A362034 od:
%p A362034 for n from 0 to 10 do
%p A362034   seq(T[n,k],k=0..n)
%p A362034 od; # _Robert Israel_, Apr 05 2023
%t A362034 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 *)
%o A362034 (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
%Y A362034 Cf. A000040, A007318, A132403, A199333.
%K A362034 nonn,tabl
%O A362034 0,1
%A A362034 _Jack Braxton_, Apr 05 2023