A220272 Number of ways to write n=x^2+y (x>0, y>0) with 2*x*y-1 prime.

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

Author

Zhi-Wei Sun, Dec 09 2012

Keywords

• nonn

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.

Examples

			a(18)=1 since 18=3^2+9 with 2*3*9-1=53 prime.
		

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.
• Zhi-Wei Sun, Re: A curious conjecture on primes, a message to Number Theory List, Dec. 12, 2012.

Crossrefs

Cf. A219842, A219864, A219923.

Programs

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}]