A284594 Numbers whose square has a prime number of partitions.

2, 6, 29, 36, 2480, 14881

Offset: 1

Author

Serge Batalov, Mar 29 2017

Keywords

• nonn
• hard
• more

Comments

Because asymptotically A072213(n) = A000041(n^2) ~ exp(Pi*sqrt(2/3)*n) / (4*sqrt(3)*n^2), the sum of the prime probabilities ~ 1/log(A072213(n)) is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.

Curiously, both A000041(6^2) and A000041(6^4) are prime; in addition, A000041(6^3) and A000041(6^1) are prime, but for no other powers A000041(6^k) is known (or can be expected) to be prime.

a(7) > 649350.

Examples

			a(2) = 6 is in the sequence because A000041(6^2) = 17977 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, A285086, A285087, A285088.

Programs

PARI

for(n=1,2500,if(ispseudoprime(numbpart(n^2)),print1(n,", ")))