A096066 - OEIS

%I #13 Sep 23 2021 05:08:38

%S 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,

%T 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,

%U 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

%N 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.

%F T(n,n) = 1.

%e n=5, A000040(5)=11 with A056768(5)=6 partitions into primes:

%e T(5,1)=10 prime(1)=2 in 7+2+2=5+2+2+2=3+3+3+2=3+2+2+2+2,

%e T(5,2)=6 prime(2)=3: in 5+3+3=3+3+3+2=3+2+2+2+2,

%e T(5,3)=2 prime(3)=5: in 5+3+3=5+2+2+2,

%e T(5,4)=1 prime(4)=7: in 7+2+2.

%e Triangle begins:

%e   1;

%e   0,  1;

%e   1,  1, 1;

%e   3,  1, 1, 1;

%e   10, 6, 2, 1, 1;

%e   ...

%t ip[p_] := ip[p] = IntegerPartitions[p, All, Select[Range[p], PrimeQ]] // Flatten;

%t T[n_, k_] := Count[ip[Prime[n]], Prime[k]];

%t Table[T[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* _Jean-François Alcover_, Sep 23 2021 *)

%Y Cf. A056768.

%K nonn,tabl

%O 1,7

%A _Reinhard Zumkeller_, Jul 21 2004

%E Name modified by _Jean-François Alcover_, Sep 23 2021