A285086 - OEIS

%I #10 Feb 16 2025 08:33:43

%S 1,2,3914

%N Numbers n such that the number of partitions of n^2+1 (=A000041(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(4) > 90000.

%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>

%e a(2) = 2 is in the sequence because A000041(2^2+1) = 7 is a prime.

%o (PARI) for(n=1,3920,if(ispseudoprime(numbpart(n^2+1)),print1(n,", ")))

%Y Cf. A000041, A046063, A072213, A284594, A285087, A285088.

%K nonn,hard,more,bref,changed

%O 1,2

%A _Serge Batalov_, Apr 09 2017