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

1,5

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, 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.

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

Cf. A219842, A219864, A219923.

Sequence in context: A173305 A233867 A319814 * A298917 A322530 A303364

Adjacent sequences: A220269 A220270 A220271 * A220273 A220274 A220275

Keyword

nonn

Author

Zhi-Wei Sun, Dec 09 2012

Status

approved