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A355704
Indices k of partition function p where p(k) and p(k) + 2 are twin primes.
4
3, 4, 6, 13, 2335166
OFFSET
1,1
COMMENTS
Because asymptotically size of partitions number function p(n) ~ O(exp(sqrt(n))), and probability of primality of p(n) ~ O(1/sqrt(n)) and 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.
a(6) > 10^7.
EXAMPLE
13 is a term because A000041(13) = 101 is prime and 103 is prime.
PROG
(PARI) for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n, ", ")))
KEYWORD
nonn,hard,more
AUTHOR
Serge Batalov, Jul 14 2022
STATUS
approved