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Indices k of partition function where consecutive p(k) and p(k+1) are prime.
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%I #14 Jul 26 2022 01:34:46

%S 2,3,4,5,1085

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

%C 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.

%C a(6) > 10^8.

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

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

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

%K nonn,hard,more

%O 1,1

%A _Serge Batalov_, Jul 15 2022