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%I #17 Nov 05 2013 11:15:22
%S 0,0,1,0,0,2,2,2,2,1,3,2,5,1,4,3,2,3,1,1,4,2,5,3,3,4,4,8,2,3,8,2,4,3,
%T 4,8,7,2,2,8,3,8,6,1,6,8,4,1,9,2,4,10,6,1,7,11,7,10,2,6,9,3,6,3,6,6,6,
%U 8,4,8,4,4,9,2,11,4,9,6,1,4,5,5,10,7,5,6,6,7,5,8,17,8,5,2,7,8,11,10,6,4
%N Number of ways to write n = x + y + z (x, y, z > 0) such that x^2 + y^2 + z^2 + z is a square, and 6*x + 1, 6*y - 1, 6*z -1 are all prime.
%C Conjecture: a(n) > 0 for all n > 5.
%H Zhi-Wei Sun, <a href="/A231168/b231168.txt">Table of n, a(n) for n = 1..5000</a>
%H Zhi-Wei Sun, <a href="http://listserv.nodak.edu/cgi-bin/wa.exe?A2=NMBRTHRY;258fe216.1311">A conjecture involving squares and primes</a>, a message to Number Theory List, Nov. 5, 2013.
%e a(19) = 1 since 19 = 13 + 5 + 1 with 13^2 + 5^2 + 1^2 + 1 = 14^2, and 6*13 + 1 = 79, 6*5 - 1 = 29, 6*1 - 1 = 5 are all prime.
%e a(444) = 1 since 444 = 76 + 28 + 340 with 76^2 + 28^2 + 340^2 + 340 = 350^2, and 6*76 + 1 = 457, 6*28 - 1 = 167, 6*340 - 1 = 2039 are all prime.
%t SQ[n_]:=IntegerQ[Sqrt[n]]
%t a[n_]:=Sum[If[PrimeQ[6i+1]&&PrimeQ[6j-1]&&PrimeQ[6(n-i-j)-1]&&SQ[i^2+j^2+(n-i-j)^2+(n-i-j)],1,0],{i,1,n-2},{j,1,n-1-i}]
%t Table[a[n],{n,1,100}]
%Y Cf. A000040, A000290, A227877, A230121, A230596, A230747.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Nov 04 2013
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