A355705 Indices k of partition function p where p(k) and p(k) - 2 are twin primes.

4, 5, 186, 3542

Offset: 1

Author

Serge Batalov, Jul 15 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(5) > 10^7.

Examples

			4 is a term because A000041(4) = 5, and 3 and 5 are twin primes.
5 is a term because A000041(5) = 7, and 5 and 7 are twin primes.
		

Crossrefs

Cf. A000041, A046063, A072213, A285086, A285087, A285088, A284594, A355704, A355706.

Programs

PARI

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