OFFSET
1,2
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
REFERENCES
Thomas Ordowski, Personal e-mail message, Nov 03 2012.
LINKS
Zhi-Wei Sun, Table of n, a(n) for n = 1..20000
EXAMPLE
For n=7 we have a(7)=1, since x^4 + (15-x)^4 with 0 < x < 8 is prime only when x=4.
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}]
PROG
(PARI) A218586(n)=sum(x=1, n+0*n=2*n+1, isprime(x^4+(n-x)^4)) \\ M. F. Hasler, Nov 05 2012
CROSSREFS
KEYWORD
nonn
AUTHOR
Zhi-Wei Sun, Nov 04 2012
STATUS
approved