A344890 Number of partitions of prime(n) containing a prime number of distinct primes and an arbitrary number of nonprimes.
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
Keywords
Examples
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).
Programs
-
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}]
Extensions
a(23)-a(33) from Alois P. Heinz, Jun 02 2021