A218654 Number of ways to write n as x+y with 0

0, 1, 1, 1, 2, 1, 2, 2, 2, 1, 4, 1, 4, 2, 4, 2, 3, 2, 6, 3, 4, 3, 6, 2, 6, 3, 4, 3, 8, 3, 8, 2, 5, 5, 8, 4, 8, 6, 5, 4, 8, 2, 10, 6, 6, 3, 11, 4, 9, 6, 9, 7, 10, 4, 14, 6, 9, 3, 11, 4, 12, 7, 9, 10, 10, 4, 11, 5, 10, 9, 15, 4, 15, 9, 9, 8, 14, 6, 12, 8, 9, 8, 18, 4, 17, 11, 9, 11, 20, 5, 14, 10, 13, 7, 16, 9, 17, 6, 16, 10

Offset: 1

Author

Zhi-Wei Sun, Nov 03 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n=2,3,4,...

It is known that any prime p = 1 or -1 (mod 5) can be written uniquely in the form x(p)^2+3x(p)y(p)+y(p)^2 with x(p)>y(p)>0.

Zhi-Wei Sun also conjectured that

(sum_{p

/(sum_{p

has the limit 1+sqrt(5) as N tends to the infinity.

These conjectures are similar to the ones mentioned in the comments in A218585.

Examples

			For n=12 we have a(12)=1 since x^2+3x(12-x)+(12-x)^2 with 0<x<=6 is prime only when x=5.
		

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
• Zhi-Wei Sun, Conjectures involving primes and quadratic forms, arXiv preprint arXiv:1211.1588, 2012.

Crossrefs

Cf. A038872, A218585.

Programs

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[x^2+3x(n-x)+(n-x)^2]==True,1,0],{x,1,n/2}]; Do[Print[n," ",a[n]],{n,1,20000}]
Table[Count[IntegerPartitions[n,{2}],?(PrimeQ[#[[1]]^2+3Times@@#+ #[[2]]^2]&)], {n,110}] (* _Harvey P. Dale, Feb 28 2018 *)

PARI

A218654(n)=sum(x=1,n\2,isprime(x^2+(2*x+n)*(n-x)))  \\ - M. F. Hasler, Nov 05 2012

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