cp's OEIS Frontend

A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.

%I A220413 #12 Jul 23 2025 01:09:09
%S A220413 1,1,1,1,1,2,1,2,1,2,2,3,3,2,3,3,3,3,2,3,4,1,4,2,3,3,3,5,5,5,3,3,5,4,
%T A220413 4,5,6,7,4,4,5,2,6,5,5,5,4,2,4,6,4,5,4,4,8,6,5,11,6,6,8,10,5,5,5,8,6,
%U A220413 6,11,7,5,7,9,7,6,7,8,9,6,8,10,7,11,8,7,10,9,9,6,5,7,8,13,7,9,13,13,12,9,9
%N A220413 Number of ways to write n=x+y (x>=0, y>=0) with x^3+2*y^3 prime.
%C A220413 Conjecture: a(n)>0 for every n=1,2,3,... Moreover, any integer n>3 not among 7, 22, 31 can be written as p+q (q>0) with p and p^3+2*q^3 both prime.
%C A220413 We have verified this conjecture for n up to 10^8. D. R. Heath-Brown proved in 2001 that there are infinitely many primes in the form x^3+2*y^3, where x and y are positive integers.
%C A220413 Zhi-Wei Sun also made the following general conjecture: For each positive odd integer m, any sufficiently large integer n can be written as x+y (x>=0, y>=0) with x^m+2*y^m prime.
%C A220413 When m=1, this follows from Bertrand's postulate proved by Chebyshev in 1850. For m = 5, 7, 9, 11, 13, 15, 17, 19, it suffices to require that n is greater than 46, 69, 141, 274, 243, 189, 320, 454 respectively.
%H A220413 Zhi-Wei Sun, <a href="/A220413/b220413.txt">Table of n, a(n) for n = 1..10000</a>
%H A220413 D. R. Heath-Brown, <a href="https://doi.org/10.1007/BF02392715">Primes represented by x^3 + 2y^3</a>. Acta Mathematica 186 (2001), pp. 1-84.
%H A220413 Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588.
%e A220413 a(9)=1 since 9=7+2 with 7^3+2*2^3=359 prime.
%e A220413 a(22)=1 since 22=1+21 with 1^3+2*21^3=18523 prime.
%t A220413 a[n_]:=a[n]=Sum[If[PrimeQ[k^3+2(n-k)^3]==True,1,0],{k,0,n}]
%t A220413 Do[Print[n," ",a[n]],{n,1,100}]
%Y A220413 Cf. A220272, A219842, A219864, A219923.
%K A220413 nonn
%O A220413 1,6
%A A220413 _Zhi-Wei Sun_, Dec 13 2012