A218585 Number of ways to write n as x+y with 0<x<=y and x^2+xy+y^2 prime.

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

1,7

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<N, p=1(mod 3)}x(p))/(sum_{p<N, p=1(mod 3)}y(p)) tends to 1+sqrt(3), and (sum_{p<N, p=1(mod 3)}x(p)^2)/(sum_{p<N, p=1(mod 3)}y(p)^2) tends to 52/9. Similar conjectures involving p=x^2+y^2 were recently formulated by Thomas Ordowski.

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

Suggestion: the number of primes of the form n*x+(n-x)^2 with 0<x<n/3 is positive for n>12. - Zak Seidov_, Sep 25 2013

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.

Example

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.

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 *)

Prog

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

Crossrefs

Cf. A002476.

Sequence in context: A318326 A329646 A293813 * A279507 A054656 A080096

Adjacent sequences: A218582 A218583 A218584 * A218586 A218587 A218588

Keyword

nonn

Author

Zhi-Wei Sun, Nov 03 2012

Status

approved