A355704 Indices k of partition function p where p(k) and p(k) + 2 are twin primes.

3, 4, 6, 13, 2335166

Offset: 1

Author

Serge Batalov, Jul 14 2022

Keywords

• nonn
• hard
• more

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.

Examples

			13 is a term because A000041(13) = 101 is prime and 103 is prime.
		

Crossrefs

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355705, A355706.

Programs

PARI

for(n=1, 2500, if(ispseudoprime(p=numbpart(n))&&ispseudoprime(p+2), print1(n,", ")))