cp's OEIS Frontend

A236442 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that A000009(k) + A047967(m) is prime.

%I A236442 #10 Apr 06 2014 10:48:36
%S A236442 0,0,1,3,3,4,3,4,3,5,2,3,4,3,1,4,4,1,2,4,4,2,4,4,3,5,8,5,4,5,7,4,3,5,
%T A236442 2,7,5,3,5,4,5,9,4,5,5,5,8,6,7,7,8,9,5,9,7,8,13,5,4,8,4,8,3,9,9,6,7,8,
%U A236442 6,9,7,7,4,10,7,6,8,8,5,9,6,10,5,10,12,6,11,5,5,9,8,8,4,4,11,8,8,12,6,8
%N A236442 Number of ordered ways to write n = k + m with k > 0 and m > 0 such that A000009(k) + A047967(m) is prime.
%C A236442 Conjecture: (i) a(n) > 0 for all n > 2.
%C A236442 (ii) If n > 2 is neither 18 nor 30, then n can be written as k + m with k > 0 and m > 0 such that A000009(k)^2 + A047967(m)^2 is prime.
%C A236442 (iii) Any integer n > 4 can be written as k + m with k > 0 and m > 0 such that A000009(k)*A047967(m) - 1 (or A000009(k)*A047967(m) + 1) is prime.
%H A236442 Zhi-Wei Sun, <a href="/A236442/b236442.txt">Table of n, a(n) for n = 1..10000</a>
%H A236442 Z.-W. Sun, <a href="http://arxiv.org/abs/1402.6641">Problems on combinatorial properties of primes</a>, arXiv:1402.6641, 2014
%e A236442 a(15) = 1 since 15 = 13 + 2 with A000009(13) + A047967(13) = 18 + 1 = 19 prime.
%e A236442 a(18) = 1 since 18 = 3 + 15 with A000009(3) + A047967(15) = 2 + 149 = 151 prime.
%t A236442 p[n_,k_]:=PrimeQ[PartitionsQ[k]+(PartitionsP[n-k]-PartitionsQ[n-k])]
%t A236442 a[n_]:=Sum[If[p[n,k],1,0],{k,1,n-1}]
%t A236442 Table[a[n],{n,1,100}]
%Y A236442 Cf. A000009, A000040, A047967, A232504, A233307, A233417, A233390.
%K A236442 nonn
%O A236442 1,4
%A A236442 _Zhi-Wei Sun_, Jan 26 2014