A344890 - OEIS

%I #17 Nov 14 2021 16:09:30

%S 0,0,1,3,20,42,151,265,753,3006,4594,15117,31576,45002,89125,235501,

%T 589613,792426,1871442,3251293,4261819,9403682,15690192,33111688,

%U 86520382,137957345,173655404,273492399,342231447,532915031,2380864800,3601147053,6628703864

%N Number of partitions of prime(n) containing a prime number of distinct primes and an arbitrary number of nonprimes.

%F a(n) = A344715(A000040(n)).

%e 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).

%p b:= proc(n, i) option remember; expand(

%p       `if`(n=0 or i=1, 1, b(n, i-1)+`if`(isprime(i), x, 1)

%p           *add(b(n-i*j, i-1), j=1..n/i)))

%p     end:

%p a:= n-> (p-> add(`if`(isprime(i), coeff(p, x, i), 0),

%p              i=2..degree(p)))(b(ithprime(n)$2)):

%p seq(a(n), n=1..33);  # _Alois P. Heinz_, Nov 14 2021

%t nterms=22;Table[Total[Map[If[PrimeQ[Count[#, _?PrimeQ]],1,0] &,Map[DeleteDuplicates[#]&,IntegerPartitions[Prime[n]],{1}]]],{n,1,nterms}]

%Y Cf. A000040, A058698, A085755, A235945, A343753, A344677, A344715.

%K nonn

%O 1,4

%A _Paolo Xausa_, Jun 01 2021

%E a(23)-a(33) from _Alois P. Heinz_, Jun 02 2021