A218656 Number of ways to write 2n+1 as x+y with 0 < x < y and x^4 + y^4 prime.

1, 2, 3, 2, 3, 3, 1, 5, 4, 4, 4, 5, 4, 7, 6, 5, 3, 10, 4, 9, 8, 4, 9, 6, 7, 11, 7, 5, 11, 9, 9, 9, 11, 4, 14, 14, 9, 8, 9, 7, 11, 8, 12, 12, 10, 9, 11, 17, 10, 12, 16, 7, 13, 14, 8, 15, 9, 11, 23, 16, 9, 17, 23, 8, 15, 15, 11, 21, 18, 12, 19, 14, 15, 19, 21, 17, 16, 23, 13, 21, 20, 17, 29

Offset: 1

Author

Zhi-Wei Sun, Nov 04 2012

Keywords

• nonn

Comments

Conjecture: a(n) > 0 for all n >= 1.

Thomas Ordowski conjectured on Nov 03 2012 that if x^4 + y^4 in the definition of a(n) is replaced by x^2 + y^2, then a(n) will always be positive.

We also have similar conjectures with x^4 + y^4 replaced by x^8 + y^8 or x^16 + y^16.

All conjectures verified for 2n+1 up to 10^6: no exceptions for x^2 + y^2 and x^4 + y^4; exceptions 2n+1 = 7, 9, 55, 73, 75 and 105 for x^8 + y^8; exceptions 2n+1 = 5 and 9 for x^16 + y^16. - Mauro Fiorentini, Sep 22 2023

Alternate definition: Number of primes of the form k^4 + (2n+1-k)^4, 0 < k <= n. - M. F. Hasler, Nov 05 2012

Examples

			For n=7 we have a(7)=1, since x^4 + (15-x)^4 with 0 < x < 8 is prime only when x=4.
		

References

• Thomas Ordowski, Personal e-mail message, Nov 03 2012.

Links

• Zhi-Wei Sun, Table of n, a(n) for n = 1..20000

Crossrefs

Cf. A002645, A218585, A218654.

Programs

Maple

A218656 := n-> add(`if`(isprime(i^4+(2*n+1-i)^4), 1, 0), i=1..n): # Alois P. Heinz, Jul 09 2016

Mathematica

a[n_]:=a[n]=Sum[If[PrimeQ[x^4+(2n+1-x)^4]==True,1,0],{x,1,n}]
Do[Print[n," ",a[n]],{n,1,20000}]

PARI

A218586(n)=sum(x=1,n+0*n=2*n+1, isprime(x^4+(n-x)^4))  \\ M. F. Hasler, Nov 05 2012