A355728 Indices k of partition function where consecutive p(k) and p(k+1) are prime.

2, 3, 4, 5, 1085

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+1) 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) > 10^8.

Examples

			5 is in the sequence because A000041(5) = 7 and A000041(6) = 11 are prime.
		

Crossrefs

Cf. A000041, A046063, A072213, A284594, A285086, A285087, A285088, A355704, A355705, A355706.

Programs

PARI

for(k=1, 5000, if(ispseudoprime(numbpart(k))&&ispseudoprime(numbpart(k+1)), print1(k, ", ")))