A230224 - OEIS

%I #7 Oct 12 2013 01:44:18

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

%T 4,5,3,6,4,6,5,5,5,7,5,9,4,6,6,8,6,9,5,7,7,10,6,8,7,8,7,9,5,10,7,11,7,

%U 7,7,11,7,10,6,10,6,13,7,9,7,11,9,11,7,9,6,14,8,12,6,13,11,12,11,13,10,16,9,14,7,14

%N Number of ways to write 2*n = p + q + r + s with p <= q <= r <= s such that p, q, r, s are primes in A230223.

%C Conjecture: a(n) > 0 for all n > 17.

%H Zhi-Wei Sun, <a href="/A230224/b230224.txt">Table of n, a(n) for n = 1..3000</a>

%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, preprint, arXiv:1211.1588.

%e a(21) = 1 since 2*21 = 7 + 7 + 11 + 17, and 7, 11, 17 are primes in A230223.

%e a(27) = 1 since 2*27 = 7 + 11 + 17 + 19, and 7, 11, 17, 19 are primes in A230223.

%t RQ[n_]:=n>5&&PrimeQ[3n-4]&&PrimeQ[3n-10]&&PrimeQ[3n-14]

%t SQ[n_]:=PrimeQ[n]&&RQ[n]

%t a[n_]:=Sum[If[RQ[Prime[i]]&&RQ[Prime[j]]&&RQ[Prime[k]]&&SQ[2n-Prime[i]-Prime[j]-Prime[k]],1,0],

%t {i,1,PrimePi[n/2]},{j,i,PrimePi[(2n-Prime[i])/3]},{k,j,PrimePi[(2n-Prime[i]-Prime[j])/2]}]

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

%Y Cf. A002375, A068307, A230223, A230140, A230141, A230219.

%K nonn

%O 1,20

%A _Zhi-Wei Sun_, Oct 12 2013