A219782 Number of ways to write n=x+y (0<x<=y) with n^2-xy and n^2+xy both prime

0, 1, 1, 1, 1, 1, 1, 0, 2, 0, 1, 1, 3, 2, 3, 1, 1, 0, 2, 0, 2, 1, 3, 1, 2, 1, 3, 2, 4, 2, 2, 1, 1, 2, 4, 2, 3, 2, 4, 3, 0, 1, 2, 2, 1, 0, 4, 1, 4, 1, 6, 2, 1, 2, 6, 1, 3, 0, 1, 3, 5, 2, 7, 2, 1, 2, 4, 1, 3, 3, 5, 2, 1, 2, 2, 2, 4, 0, 3, 1, 5, 2, 4, 3, 2, 3, 2, 3, 2, 1, 4, 3, 3, 2, 3, 2, 7, 1, 5, 5

Offset

1,9

Comments

Conjecture: a(n)>0 if n is not among 1, 8, 10, 18, 20, 41, 46, 58, 78, 116, 440.

Zhi-Wei Sun also made the following general conjecture:

For any k=0,1,2,4,5,6,... and positive odd integer m, each sufficiently large integer n can be written as x+y (0<x<=y) with |m*n^k-xy| and |m*n^k+xy| all prime.

For example, if n>6 is different from 24 then n can be written as x+y with x,y positive, and xy-n and xy+n both prime; if n>308 then n can be written as x+y with x,y positive, and 3n^2-xy and 3n^2+xy both prime.

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(9)=2 since 9=1+8=4+5 with 9^2+1*8, 9^2-1*8, 9^2+4*5, 9^2-4*5 all prime.

Mathematica

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

Crossrefs

Cf. A091182, A218585, A218654.

Sequence in context: A372442 A107261 A265336 * A035788 A097559 A293108

Adjacent sequences: A219779 A219780 A219781 * A219783 A219784 A219785

Keyword

nonn

Author

Zhi-Wei Sun, Nov 27 2012

Status

approved