A219782 Number of ways to write n=x+y (0

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

Author

Zhi-Wei Sun, Nov 27 2012

Keywords

• nonn

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

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.

Examples

			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.
		

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.

Crossrefs

Cf. A091182, A218585, A218654.

Programs

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}]

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.