A284594 - OEIS

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A284594

Numbers whose square has a prime number of partitions.

2, 6, 29, 36, 2480, 14881

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OFFSET

1,1

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.

LINKS

Table of n, a(n) for n=1..6.

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

EXAMPLE

a(2) = 6 is in the sequence because A000041(6^2) = 17977 is a prime.

PROG

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

CROSSREFS

Cf. A000041, A046063, A072213, A285086, A285087, A285088.