%I #17 Jul 26 2022 01:34:35
%S 4,5,186,3542
%N Indices k of partition function p where p(k) and p(k) - 2 are twin primes.
%C 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.
%C a(5) > 10^7.
%e 4 is a term because A000041(4) = 5, and 3 and 5 are twin primes.
%e 5 is a term because A000041(5) = 7, and 5 and 7 are twin primes.
%o (PARI) for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p-2), print1(n, ", ")))
%Y Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355704, A355706.
%K nonn,hard,more
%O 1,1
%A _Serge Batalov_, Jul 15 2022