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URL: https://oeis.org/A344890

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A344890
Number of partitions of prime(n) containing a prime number of distinct primes and an arbitrary number of nonprimes.
1
0, 0, 1, 3, 20, 42, 151, 265, 753, 3006, 4594, 15117, 31576, 45002, 89125, 235501, 589613, 792426, 1871442, 3251293, 4261819, 9403682, 15690192, 33111688, 86520382, 137957345, 173655404, 273492399, 342231447, 532915031, 2380864800, 3601147053, 6628703864
OFFSET
1,4
FORMULA
a(n) = A344715(A000040(n)).
EXAMPLE
a(4) = 3 because there are 3 partitions of prime(4)=7 that contain a prime number of primes (not counting repetitions). These partitions are [5,2] (containing 2 primes), [3,2,2] (containing 2 unique primes) and [3,2,1,1] (containing 2 primes).
MAPLE
b:= proc(n, i) option remember; expand(
`if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1)
*add(b(n-i*j, i-1), j=1..n/i)))
end:
a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0),
i=2..degree(p)))(b(ithprime(n)$2)):
seq(a(n), n=1..33); # Alois P. Heinz, Nov 14 2021
MATHEMATICA
nterms=22; Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]], 1, 0] &, Map[DeleteDuplicates[#]&, IntegerPartitions[Prime[n]], {1}]]], {n, 1, nterms}]
KEYWORD
nonn
AUTHOR
Paolo Xausa, Jun 01 2021
EXTENSIONS
a(23)-a(33) from Alois P. Heinz, Jun 02 2021
STATUS
approved