A285087 - OEIS

%I #20 Feb 16 2025 08:33:43

%S 2,13,21,46909

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

%C 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.

%C a(5) > 50000.

%H Chris K. Caldwell, <a href="https://t5k.org/top20/page.php?id=54">Top twenty prime partition numbers</a>, The Prime Pages.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PartitionFunctionP.html">Partition Function P</a>

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/IntegerSequencePrimes.html">Integer Sequence Primes</a>

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

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

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

%o (Python)

%o from itertools import count, islice

%o from sympy import isprime, npartitions

%o def A285087_gen(startvalue=1): # generator of terms >= startvalue

%o     return filter(lambda n: isprime(npartitions(n**2-1)), count(max(startvalue,1)))

%o A285087_list = list(islice(A285087_gen(),3)) # _Chai Wah Wu_, Nov 20 2023

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

%K nonn,hard,more,changed

%O 1,1

%A _Serge Batalov_, Apr 09 2017