cp's OEIS Frontend

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

%I A238458 #10 Apr 06 2014 22:32:05
%S A238458 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 A238458 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 A238458 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 A238458 Number of primes p < n with 2*P(n-p) + 1 prime, where P(.) is the partition function (A000041).
%C A238458 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 A238458 We have verified the conjecture for n up to 10^5.
%C A238458 See also A238459 for a similar conjecture involving the strict partition function.
%H A238458 Zhi-Wei Sun, <a href="/A238458/b238458.txt">Table of n, a(n) for n = 1..10000</a>
%H A238458 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A238458 a(3) = 1 since 2 and 2*P(3-2) + 1 = 2*1 + 1 = 3 are both prime.
%e A238458 a(41) = 1 since 37 and 2*P(41-37) + 1 = 2*5 + 1 = 11 are both prime.
%t A238458 p[n_,k_]:=PrimeQ[2*PartitionsP[n-Prime[k]]+1]
%t A238458 a[n_]:=Sum[If[p[n,k],1,0],{k,1,PrimePi[n-1]}]
%t A238458 Table[a[n],{n,1,100}]
%Y A238458 Cf. A000040, A000041, A237705, A237768, A237769, A238457, A238459.
%K A238458 nonn
%O A238458 1,4
%A A238458 _Zhi-Wei Sun_, Feb 27 2014