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A343813 Number of partitions of prime(n) containing at least one prime.

%I A343813 #29 May 26 2021 13:47:29
%S A343813 1,2,5,12,48,88,269,450,1176,4355,6558,20958,43412,61733,122194,
%T A343813 324532,820827,1107647,2652517,4655220,6133664,13751210,23192039,
%U A343813 49730098,132657130,213646624,270244858,429702432,540212859,848899870,3905568236,5952945182,11078643138
%N A343813 Number of partitions of prime(n) containing at least one prime.
%H A343813 David A. Corneth, <a href="/A343813/b343813.txt">Table of n, a(n) for n = 1..10000</a>
%F A343813 a(n) = A235945(A000040(n)).
%e A343813 a(4) = 12 because there are 12 partitions of prime(4) = 7 that contain at least one prime. These partitions are [7], [5,2], [5,1,1], [4,3], [4,2,1], [3,3,1], [3,2,2], [3,2,1,1], [3,1,1,1,1], [2,2,2,1], [2,2,1,1,1], [2,1,1,1,1,1].
%t A343813 nterms=20;Table[Total[Map[If[Count[#, _?PrimeQ]>0,1,0] &,IntegerPartitions[Prime[n]]]],{n,1,nterms}]
%o A343813 (PARI) forprime(p=2,59,my(m=0); forpart(X=p, for(k=1,#X, if(isprime(X[k]),m++;break))); print1(m,", ")) \\ _Hugo Pfoertner_, Apr 30 2021
%o A343813 (PARI) seq(n)={my(p=primes(n), m=p[#p]); vecextract(Vec(1/eta(x+O(x*x^m)) - 1/prod(k=1, m, 1-if(!isprime(k), x^k) + O(x*x^m)), -m), p)} \\ _Andrew Howroyd_, Apr 30 2021
%o A343813 (Python)
%o A343813 from sympy.utilities.iterables import partitions
%o A343813 from sympy import sieve, prime
%o A343813 def A343813(n):
%o A343813     p = prime(n)
%o A343813     pset = set(sieve.primerange(2,p+1))
%o A343813     return sum(1 for d in partitions(p) if len(set(d)&pset) > 0) # _Chai Wah Wu_, May 01 2021
%Y A343813 Cf. A000040, A058698, A235945.
%K A343813 nonn
%O A343813 1,2
%A A343813 _Paolo Xausa_, Apr 30 2021