OFFSET
1,14
COMMENTS
Conjecture: a(n)>0 for all n>5.
We have verified this for n up to 10^9. It is stronger than Goldbach's conjecture and Lemoine's conjecture.
Zhi-Wei Sun also conjectured the following refinement: Any odd number 2n+1>64 not among 105, 247, 255, 1105 can be written as p+2q, where p and q are primes, and JacobiSymbol[q,p']=1 for any prime divisor p' of 2n+1; also, any even number 2n>8 not among 32 and 152 can be written as p+q, where p and q<=n/2 are primes, and JacobiSymbol[(q+1)/2,p']=1 for any prime divisor p' of 2n+1.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.
EXAMPLE
a(19)=1 since 19=5+2*7 with JacobiSymbol[7,19]=1.
a(32)=1 since 32=29+3 with JacobiSymbol[(3+1)/2,32+1]=1.
MATHEMATICA
a[n_]:=a[n]=Sum[If[(Mod[n, 2]==1&&PrimeQ[n-2Prime[k]]==True&&JacobiSymbol[Prime[k], n]==1)||(Mod[n, 2]==0&&n-Prime[k]>2&&PrimeQ[n-Prime[k]]==True&&JacobiSymbol[(Prime[k]+1)/2, n+1]==1), 1, 0], {k, 1, PrimePi[n/2]}]
Do[Print[n, " ", a[n]], {n, 1, 200}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 30 2012
STATUS
approved