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 A231168 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. 1
 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, 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, 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 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,6 COMMENTS Conjecture: a(n) > 0 for all n > 5. LINKS Zhi-Wei Sun, Table of n, a(n) for n = 1..5000 Zhi-Wei Sun, A conjecture involving squares and primes, a message to Number Theory List, Nov. 5, 2013. EXAMPLE 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. 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. MATHEMATICA SQ[n_]:=IntegerQ[Sqrt[n]] 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}] Table[a[n], {n, 1, 100}] CROSSREFS Cf. A000040, A000290, A227877, A230121, A230596, A230747. Sequence in context: A306242 A118144 A136691 * A276830 A064741 A281659 Adjacent sequences:  A231165 A231166 A231167 * A231169 A231170 A231171 KEYWORD nonn AUTHOR Zhi-Wei Sun, Nov 04 2013 STATUS approved

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Last modified June 26 00:10 EDT 2019. Contains 324367 sequences. (Running on oeis4.)