cp's OEIS Frontend

A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.

%I A231633 #29 Oct 19 2023 22:50:12
%S A231633 0,0,1,2,3,1,3,2,5,2,4,2,7,2,5,3,5,3,10,4,5,3,8,3,14,6,5,4,11,5,11,3,
%T A231633 11,9,4,5,10,5,11,9,12,3,19,7,11,6,12,9,11,7,17,7,13,5,22,3,3,15,16,5,
%U A231633 25,4,9,11,13,5,19,6,22,6,11,6,39,6,24,7,7,6,25,8,21,11,24,7,31,7,19,11,33,10,14,8,15,27,18,9,21,4,27,9
%N A231633 Number of ways to write n = x + y (x, y > 0) with x^2 * y - 1 prime.
%C A231633 Conjectures:
%C A231633 (i) a(n) > 0 for all n > 2. Also, any integer n > 4 can be written as x + y (x, y > 0) with x^2 * y + 1 prime.
%C A231633 (ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with (x*y)^2 + x*y + 1 prime.
%C A231633 (iii) Also, any integer n > 2 can be written as x + y (x, y > 0) with 2*(x*y)^2 - 1 (or (x*y)^2 + x*y - 1) prime.
%C A231633 From _Mauro Fiorentini_, Jul 31 2023: (Start)
%C A231633 Both parts of conjecture (i) verified for n up to 10^9.
%C A231633 Conjecture (ii) and both parts of conjecture (iii) verified for n up to 10^7. (End)
%H A231633 Zhi-Wei Sun, <a href="/A231633/b231633.txt">Table of n, a(n) for n = 1..10000</a>
%e A231633 a(6) = 1 since 6 = 4 + 2 with 4^2*2 - 1 = 31 prime.
%t A231633 a[n_]:=Sum[If[PrimeQ[x^2*(n-x)-1],1,0],{x,1,n-1}]
%t A231633 Table[a[n],{n,1,100}]
%Y A231633 Cf. A000040, A000290, A109909, A219842, A219864, A220272, A228424, A231201, A231155, A231561, A231557, A231631.
%K A231633 nonn
%O A231633 1,4
%A A231633 _Zhi-Wei Sun_, Nov 12 2013