login
A230514
Number of ways to write n = a + b + c (0 < a <= b <= c) such that all the three numbers a*(a+1)-1, b*(b+1)-1, c*(c+1)-1 are Sophie Germain primes.
3
0, 0, 0, 0, 0, 1, 1, 1, 2, 2, 2, 2, 3, 3, 4, 3, 4, 3, 4, 4, 4, 3, 5, 4, 4, 4, 5, 4, 4, 2, 4, 4, 4, 2, 3, 2, 3, 2, 1, 2, 2, 3, 3, 3, 4, 5, 3, 2, 5, 6, 5, 5, 6, 5, 7, 9, 6, 7, 9, 9, 8, 10, 8, 8, 10, 7, 8, 10, 6, 9, 8, 6, 5, 8, 4, 7, 4, 4, 8, 7, 5, 3, 5, 3, 7, 3, 3, 5, 7, 5, 4, 6, 5, 6, 7, 5, 6, 10, 9, 6
OFFSET
1,9
COMMENTS
Conjecture: a(n) > 0 for all n > 5.
Conjecture verified for n up to 10^9. - Mauro Fiorentini, Sep 22 2023
This implies that there are infinitely many Sophie Germain primes of the form x^2 + x - 1.
See also A230516 for a similar conjecture.
LINKS
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, preprint, arXiv:1211.1588 [math.NT], 2012-2017.
EXAMPLE
a(10) = 2 since 10 = 2 + 2 + 6 = 2 + 3 + 5, and 2*3 - 1 = 5, 6*7 - 1 = 41, 3*4 - 1 = 11, 5*6 - 1 = 29 are all Sophie Germain primes.
a(39) = 1 since 39 = 9 + 15 + 15, and both 9*10 - 1 = 89 and 15*16 - 1 = 239 are Sophie Germain primes.
MATHEMATICA
pp[n_]:=PrimeQ[n(n+1)-1]&&PrimeQ[2n(n+1)-1]
a[n_]:=Sum[If[pp[i]&&pp[j]&&pp[n-i-j], 1, 0], {i, 1, n/3}, {j, i, (n-i)/2}]
Table[a[n], {n, 1, 100}]
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Oct 21 2013
STATUS
approved