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

1,7

Links

Table of n, a(n) for n=1..91.

Formula

T(n,n) = 1.

Example

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;
  ...

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 *)

Crossrefs

Cf. A056768.

Sequence in context: A080214 A263383 A185620 * A294746 A326180 A064085

Adjacent sequences: A096063 A096064 A096065 * A096067 A096068 A096069

Keyword

nonn,tabl

Author

Reinhard Zumkeller, Jul 21 2004

Extensions

Name modified by Jean-François Alcover, Sep 23 2021

Status

approved