A219923 Number of ways to write n=x+y (x>0, y>0) with x-1, x+1 and 2*x*y+1 all prime

0, 0, 0, 0, 0, 1, 1, 0, 2, 0, 1, 1, 1, 1, 3, 2, 0, 1, 2, 2, 3, 2, 1, 0, 2, 2, 0, 1, 3, 2, 2, 1, 3, 4, 2, 2, 3, 0, 4, 3, 3, 1, 1, 3, 0, 3, 2, 1, 1, 3, 3, 1, 1, 5, 3, 1, 2, 1, 3, 3, 5, 3, 1, 2, 4, 3, 3, 2, 4, 3, 2, 2, 0, 3, 5, 4, 1, 3, 6, 2, 6, 2, 2, 4, 5, 5, 2, 3, 3, 4, 1, 2, 0, 1, 4, 2, 4, 1, 6, 6

Offset

1,9

Comments

Conjecture: a(n)>0 for all n>623.

This has been verified for n up to 10^8.

Zhi-Wei Sun made the following general conjecture: For each nonnegative integer m, any sufficiently large integer n can be written as x+y (x>0, y>0) with x-m, x+m and 2*x*y+1 all prime.

For example, when m = 2, 3, 4, 5 it suffices to require that n is greater than 28, 151, 357, 199 respectively.

Sun also conjectured that for each m=0,1,2,... any sufficiently large integer n with m or n odd can be written as x+y (x>0, y>0) with x-m, x+m and x*y-1 all prime.

For example, in the case m=1 it suffices to require that n is greater than 4 and not among 40, 125, 155, 180, 470, 1275, 2185, 3875; when m=2 it suffices to require that n is odd, greater than 7, and different from 13.

Links

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv:1211.1588.

Example

a(11)=1 since 11=6+5 with 6-1, 6+1 and 2*6*5+1=61 all prime.

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[Prime[k]+2]==True&&PrimeQ[2(Prime[k]+1)(n-Prime[k]-1)+1]==True, 1, 0], {k, 1, PrimePi[n-1]}]
Do[Print[n, " ", a[n]], {n, 1, 10000}]

Crossrefs

Cf. A219864, A219842.

Sequence in context: A029425 A219055 A025902 * A286950 A210638 A272903

Adjacent sequences: A219920 A219921 A219922 * A219924 A219925 A219926

Keyword

nonn

Author

Zhi-Wei Sun, Dec 01 2012

Status

approved