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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.
0
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
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
KEYWORD
nonn,tabl
AUTHOR
Reinhard Zumkeller, Jul 21 2004
EXTENSIONS
Name modified by Jean-François Alcover, Sep 23 2021
STATUS
approved