A218585 Number of ways to write n as x+y with 0

0, 1, 1, 1, 1, 1, 2, 0, 3, 1, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 4, 3, 4, 2, 3, 3, 3, 3, 5, 2, 6, 2, 4, 4, 5, 3, 5, 2, 8, 4, 4, 4, 7, 3, 5, 2, 8, 4, 7, 2, 8, 4, 7, 5, 7, 4, 7, 3, 8, 4, 9, 3, 11, 4, 8, 5, 10, 4, 9, 5, 9, 6, 8, 5, 6, 6, 10, 5, 10, 3, 12, 7, 10, 6, 8, 6, 11, 4, 7, 4, 15, 8, 13, 6, 9, 5, 15, 9, 10

Offset: 1

Author

Zhi-Wei Sun, Nov 03 2012

Keywords

• nonn

Comments

Conjecture: a(n)>0 for all n>1 with the only exception n=8.

Note that any prime p=1(mod 3) can be written uniquely in the form x(p)^2+x(p)y(p)+y(p)^2 with x(p)>y(p)>0.

Zhi-Wei Sun also conjectured that

(sum_{p

Or, the number of primes of the form n*x+(n-x)^2 with 0

Suggestion: the number of primes of the form n*x+(n-x)^2 with 012. - Zak Seidov_, Sep 25 2013

Examples

			For n=20 we have a(20)=1 since x^2+x(20-x)+(20-x)^2 with 0<x<=10 is prime only when x=3.
		

References

• Thomas Ordowski, Personal e-mail messages, Oct. 3-4, 2012, and Nov. 3, 2012.

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. A002476.

Programs

Mathematica

A[n_]:=A[n]=Sum[If[PrimeQ[x^2+x(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+#[[1]]#[[2]]+ #[[2]]^2]&)],{n,100}] (* _Harvey P. Dale, Aug 04 2020 *)

PARI

A218585(n)=sum(x=1,n\2,isprime(x^2+x*(n-x)+(n-x)^2))  \\ - M. F. Hasler, Nov 03 2012

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