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A355728
Indices k of partition function where consecutive p(k) and p(k+1) are prime.
0
2, 3, 4, 5, 1085
OFFSET
1,1
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.
EXAMPLE
5 is in the sequence because A000041(5) = 7 and A000041(6) = 11 are prime.
PROG
(PARI) for(k=1, 5000, if(ispseudoprime(numbpart(k))&&ispseudoprime(numbpart(k+1)), print1(k, ", ")))
KEYWORD
nonn,hard,more
AUTHOR
Serge Batalov, Jul 15 2022
STATUS
approved