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A238458 Number of primes p < n with 2*P(n-p) + 1 prime, where P(.) is the partition function (A000041). 3

%I

%S 0,0,1,2,2,3,2,3,4,2,3,3,3,5,2,4,4,5,4,5,4,4,3,3,3,4,4,4,2,4,2,5,4,4,

%T 5,3,3,6,3,4,1,3,4,7,6,4,4,4,4,4,4,5,3,5,5,7,3,3,4,6,5,8,5,5,4,4,2,7,

%U 5,7,3,6,5,7,6,7,5,5,4,7,4,5,3,5,6,8,5,3,4,6,3,5,4,5,4,5,2,6,4,5

%N Number of primes p < n with 2*P(n-p) + 1 prime, where P(.) is the partition function (A000041).

%C Conjecture: a(n) > 0 for all n > 2. Also, for each n > 3 there is a prime p < n with 2*P(n-p) - 1 prime.

%C We have verified the conjecture for n up to 10^5.

%C See also A238459 for a similar conjecture involving the strict partition function.

%H Zhi-Wei Sun, <a href="/A238458/b238458.txt">Table of n, a(n) for n = 1..10000</a>

%H Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014

%e a(3) = 1 since 2 and 2*P(3-2) + 1 = 2*1 + 1 = 3 are both prime.

%e a(41) = 1 since 37 and 2*P(41-37) + 1 = 2*5 + 1 = 11 are both prime.

%t p[n_,k_]:=PrimeQ[2*PartitionsP[n-Prime[k]]+1]

%t a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]

%t Table[a[n],{n,1,100}]

%Y Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.

%K nonn

%O 1,4

%A _Zhi-Wei Sun_, Feb 27 2014

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Last modified April 5 00:43 EDT 2020. Contains 333238 sequences. (Running on oeis4.)