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
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 04 2013
STATUS
approved