OFFSET
1,1
COMMENTS
Because asymptotically the size of the partition number function p(n) is ~ O(exp(sqrt(n))), and the probability of primality of p(n) is ~ O(1/sqrt(n)) and the combined probability of primality of p(n) and p(n)+-2 is ~ O(1/n), the sum of the prime probabilities is diverging and there are no obvious restrictions on primality; therefore, this sequence may be conjectured to be infinite.
p(2335166), a 1696-digit number, was known to be prime and proven prime by F. Morain using his software (ca. April 2001), but the primality of p(2335166)+2 was found by targeted search (for this sequence) in July 2022.
a(9) > 10^7.
EXAMPLE
13 is a term because A000041(13) = 101 is prime and 101 and 103 are twin primes.
PROG
(PARI) for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))
CROSSREFS
KEYWORD
nonn,hard,more
AUTHOR
Serge Batalov, Jul 15 2022
STATUS
approved