cp's OEIS Frontend

A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.

%I A220572 #14 Jul 23 2025 01:14:11
%S A220572 1,2,1,1,1,1,2,1,1,2,1,4,5,4,1,2,4,1,4,1,2,1,2,1,6,1,4,2,4,3,6,3,2,4,
%T A220572 2,5,6,4,5,4,5,5,8,7,4,7,7,6,7,4,6,7,5,6,3,11,7,1,5,3,5,6,6,10,4,13,
%U A220572 12,9,4,9,10,5,8,3,6,7,5,4,8,13,6,3,5,5,11,6,13,4,9,10,8,12,11,8,7,10,8,7,8,8
%N A220572 Number of ways to write 2n-1=x+y (x,y>=0) with x^18+3*y^18 prime.
%C A220572 Conjecture: a(n)>0 for every n=1,2,3,.... Moreover, any odd integer greater than 2092 can be written as x+y (x,y>0) with x-3, x+3 and x^18+3*y^18 all prime.
%C A220572 This has been verified for n up to 2*10^6.
%C A220572 Zhi-Wei Sun also made the following general conjecture: For each positive integer m, any sufficiently large odd integer n can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime (and hence there are infinitely many primes in the form x^m+3*y^m). In particular, for m = 1, 2, 3, 4, 5, 6, 18 any odd integer greater than one can be written as x+y (x,y>0) with x^m+3*y^m prime, and for m =1, 2, 3 any odd integer n>15 can be written as x+y (x,y>0) with x-3, x+3 and x^m+3*y^m all prime.
%C A220572 Our computation suggests that for each m=7,...,20 any odd integer greater than 32, 10, 24, 30, 48, 36, 72, 146, 48, 48, 152, 2, 238, 84 respectively can be written as x+y (x,y>0) with x^m+3*y^m prime.
%H A220572 Zhi-Wei Sun, <a href="/A220572/b220572.txt">Table of n, a(n) for n = 1..5000</a>
%H A220572 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A220572 a(3)=1 since 2*3-1=5=1+4 with 1^18+3*4^18=206158430209 prime.
%t A220572 a[n_]:=a[n]=Sum[If[PrimeQ[k^18+3*(2n-1-k)^18]==True,1,0],{k,0,2n-1}]
%t A220572 Do[Print[n," ",a[n]],{n,1,100}]
%Y A220572 Cf. A220554, A036468, A220455, A220431, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.
%K A220572 nonn
%O A220572 1,2
%A A220572 _Zhi-Wei Sun_, Dec 16 2012