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

1, 2, 5, 12, 48, 88, 269, 450, 1176, 4355, 6558, 20958, 43412, 61733, 122194, 324532, 820827, 1107647, 2652517, 4655220, 6133664, 13751210, 23192039, 49730098, 132657130, 213646624, 270244858, 429702432, 540212859, 848899870, 3905568236, 5952945182, 11078643138

Offset

1,2

Links

David A. Corneth, Table of n, a(n) for n = 1..10000

Formula

a(n) = A235945(A000040(n)).

Example

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].

Mathematica

nterms=20; Table[Total[Map[If[Count[#, _?PrimeQ]>0, 1, 0] &, IntegerPartitions[Prime[n]]]], {n, 1, nterms}]

Prog

(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
(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
(Python)
from sympy.utilities.iterables import partitions
from sympy import sieve, prime
def A343813(n):
    p = prime(n)
    pset = set(sieve.primerange(2, p+1))
    return sum(1 for d in partitions(p) if len(set(d)&pset) > 0) # Chai Wah Wu, May 01 2021

Crossrefs

Cf. A000040, A058698, A235945.

Sequence in context: A183758 A334811 A071787 * A332791 A291484 A145997

Adjacent sequences: A343810 A343811 A343812 * A343814 A343815 A343816

Keyword

nonn

Author

Paolo Xausa, Apr 30 2021

Status

approved