login
A220272
Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime
13
0, 0, 1, 1, 2, 1, 2, 1, 1, 3, 2, 2, 2, 1, 1, 3, 3, 1, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 1, 1, 4, 3, 1, 2, 2, 2, 5, 3, 3, 3, 4, 3, 3, 1, 3, 3, 2, 2, 4, 4, 2, 6, 2, 2, 4, 4, 2, 3, 1, 2, 5, 4, 1, 3, 3, 3, 6, 2, 3, 5, 4, 3, 3, 3, 3, 6, 3, 2, 4, 2, 3, 4, 3, 2, 5, 3, 5, 2, 1, 1, 9, 4, 3, 4, 3, 5, 3, 2, 2, 2
OFFSET
1,5
COMMENTS
Conjecture: a(n)>0 for all n>2.
This has been verified for n up to 3*10^9. The author observed that for each n=3,...,3*10^9 we may even require x<(log n)^2, but Jack Brennen found that for n=4630581798 we cannot require x<(log n)^2.
The author guessed that the conjecture can be slightly refined as follows: Any integer n>2 can be written as x^2+y with 2*x*y-1 prime, where x and y are positive integers with x<=y.
Zhi-Wei Sun also made the following general conjecture: If m is a positive integer and r is 1 or -1, then any sufficiently large integer n can be written as x^2+y (x>0, y>0) with m*x*y+r prime.
For example, for (m,r)=(1,-1),(1,1),(2,1),(3,-1),(3,1),(4,-1),(4,1),(5,-1),(5,1),(6,-1),(6,1), it suffices to require that n is greater than 12782, 15372, 488, 5948, 2558, 92, 822, 21702, 6164, 777, 952 respectively.
LINKS
Zhi-Wei Sun, Re: A curious conjecture on primes, a message to Number Theory List, Dec. 12, 2012.
EXAMPLE
a(18)=1 since 18=3^2+9 with 2*3*9-1=53 prime.
MATHEMATICA
a[n_]:=a[n]=Sum[If[PrimeQ[2k(n-k^2)-1]==True, 1, 0], {k, 1, Sqrt[n]}]
Do[Print[n, " ", a[n]], {n, 1, 100}]
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Dec 09 2012
STATUS
approved