A355706 Indices k of partition function p where p(k) is a twin prime.

3, 4, 5, 6, 13, 186, 3542, 2335166

Offset: 1

Author

Serge Batalov, Jul 15 2022

Keywords

• nonn
• hard
• more

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.

Examples

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

Crossrefs

Subsequence of A046063.

Union of A355704 and A355705.

Cf. A000041, A072213, A285086, A285087, A285088, A284594.

Programs

PARI

for(n=1, 3600, if(ispseudoprime(p=numbpart(n))&&(ispseudoprime(p-2)||ispseudoprime(p+2)), print1(n, ", ")))