A285087 Numbers n such that the number of partitions of n^2-1 is prime.

2, 13, 21, 46909

Offset: 1

Author

Serge Batalov, Apr 09 2017

Keywords

• nonn
• hard
• more

Comments

Because asymptotically A000041(n^2-1) ~ exp(Pi*sqrt(2/3*(n^2-1))) / (4*sqrt(3)*(n^2-1)), the sum of the prime probabilities ~1/log(A000041(n^2-1)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

a(5) > 50000.

Examples

			13 is in the sequence because A000041(13^2-1) = 228204732751 is a prime.
		

Links

• Chris K. Caldwell, Top twenty prime partition numbers, The Prime Pages.
• Eric Weisstein's World of Mathematics, Partition Function P
• Eric Weisstein's World of Mathematics, Integer Sequence Primes

Crossrefs

Cf. A000041, A046063, A072213, A284594, A285086, A285088.

Programs

PARI

for(n=1,2000,if(ispseudoprime(numbpart(n^2-1)),print1(n,", ")))

Python

from itertools import count, islice
from sympy import isprime, npartitions
def A285087_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n: isprime(npartitions(n**2-1)), count(max(startvalue,1)))
A285087_list = list(islice(A285087_gen(),3)) # Chai Wah Wu, Nov 20 2023

Formula

{n: A000041(n^2-1) in A000040}.