A096066 Triangle read by rows, 1<=k<=n: T(n,k) is the number of occurrences of the k-th prime in partitions of the n-th prime into primes.

1, 0, 1, 1, 1, 1, 3, 1, 1, 1, 10, 6, 2, 1, 1, 16, 9, 4, 2, 1, 1, 37, 22, 11, 6, 2, 1, 1, 54, 32, 15, 9, 3, 2, 1, 1, 107, 65, 32, 19, 7, 5, 2, 1, 1, 266, 165, 84, 50, 22, 15, 7, 5, 2, 1, 353, 219, 112, 69, 30, 21, 10, 7, 3, 1, 1, 779, 487, 254, 157, 73, 52, 27, 19, 10, 3, 2, 1, 1270, 795, 420, 261, 124, 90, 49, 36, 19, 7, 5, 1, 1

Offset: 1

Author

Reinhard Zumkeller, Jul 21 2004

Keywords

• nonn
• tabl

Examples

			n=5, A000040(5)=11 with A056768(5)=6 partitions into primes:
T(5,1)=10 prime(1)=2 in 7+2+2=5+2+2+2=3+3+3+2=3+2+2+2+2,
T(5,2)=6 prime(2)=3: in 5+3+3=3+3+3+2=3+2+2+2+2,
T(5,3)=2 prime(3)=5: in 5+3+3=5+2+2+2,
T(5,4)=1 prime(4)=7: in 7+2+2.
Triangle begins:
  1;
  0,  1;
  1,  1, 1;
  3,  1, 1, 1;
  10, 6, 2, 1, 1;
  ...
		

Crossrefs

Cf. A056768.

Programs

Mathematica

ip[p_] := ip[p] = IntegerPartitions[p, All, Select[Range[p], PrimeQ]] // Flatten;
T[n_, k_] := Count[ip[Prime[n]], Prime[k]];
Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 23 2021 *)

Formula

T(n,n) = 1.

Extensions

Name modified by Jean-François Alcover, Sep 23 2021