This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A218656 #34 Dec 21 2023 11:19:22
%S A218656 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,
%T A218656 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,
%U A218656 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
%N A218656 Number of ways to write 2n+1 as x+y with 0 < x < y and x^4 + y^4 prime.
%C A218656 Conjecture: a(n) > 0 for all n >= 1.
%C A218656 _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.
%C A218656 We also have similar conjectures with x^4 + y^4 replaced by x^8 + y^8 or x^16 + y^16.
%C A218656 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
%C A218656 Alternate definition: Number of primes of the form k^4 + (2n+1-k)^4, 0 < k <= n. - _M. F. Hasler_, Nov 05 2012
%D A218656 Thomas Ordowski, Personal e-mail message, Nov 03 2012.
%H A218656 Zhi-Wei Sun, <a href="/A218656/b218656.txt">Table of n, a(n) for n = 1..20000</a>
%e A218656 For n=7 we have a(7)=1, since x^4 + (15-x)^4 with 0 < x < 8 is prime only when x=4.
%p A218656 A218656 := n-> add(`if`(isprime(i^4+(2*n+1-i)^4), 1, 0), i=1..n): # _Alois P. Heinz_, Jul 09 2016
%t A218656 a[n_]:=a[n]=Sum[If[PrimeQ[x^4+(2n+1-x)^4]==True,1,0],{x,1,n}]
%t A218656 Do[Print[n," ",a[n]],{n,1,20000}]
%o A218656 (PARI) A218586(n)=sum(x=1,n+0*n=2*n+1, isprime(x^4+(n-x)^4)) \\ _M. F. Hasler_, Nov 05 2012
%Y A218656 Cf. A002645, A218585, A218654.
%K A218656 nonn
%O A218656 1,2
%A A218656 _Zhi-Wei Sun_, Nov 04 2012