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 A231635 #18 Aug 02 2023 10:02:19
%S A231635 0,1,1,1,2,1,3,2,1,1,4,2,6,1,3,2,8,4,5,4,3,2,7,5,6,2,3,2,8,5,10,6,3,1,
%T A231635 8,3,9,4,4,4,14,6,16,7,7,2,12,6,8,4,5,5,21,5,8,6,4,8,11,7,12,5,6,4,10,
%U A231635 8,22,6,10,6,17,9,23,7,11,12,18,10,19,10,10,7,23,8,15,4,7,8,14,11,19,9,2,4,11,10,35,6,10,10
%N A231635 Number of ways to write n = x + y with 0 < x <= y such that lcm(x, y) + 1 is prime.
%C A231635 Conjecture: (i) a(n) > 0 for all n > 1. Also, any integer n > 3 can be written as x + y (x, y > 0) with lcm(x, y) - 1 prime.
%C A231635 (ii) Each n = 2, 3, ... can be expressed as x + y (x, y > 0) with lcm(x, y)^2 + lcm(x, y) + 1 prime. Also, any integer n > 1 not equal to 10 can be written as x + y (x, y > 0) with lcm(x, y)^2 + 1 prime.
%C A231635 From _Mauro Fiorentini_, Aug 02 2023: (Start)
%C A231635 Both parts of conjecture (i) verified for n up to 10^9.
%C A231635 Both parts of conjecture (ii) verified for n up to 10^6. (End)
%H A231635 Zhi-Wei Sun, <a href="/A231635/b231635.txt">Table of n, a(n) for n = 1..10000</a>
%e A231635 a(9) = 1 since 9 = 3 + 6 with lcm(3, 6) + 1 = 7 prime.
%e A231635 a(10) = 1 since 10 = 4 + 6 with lcm(4, 6) + 1 = 13 prime.
%t A231635 a[n_]:=Sum[If[PrimeQ[LCM[x,n-x]+1],1,0],{x,1,n/2}]
%t A231635 Table[a[n],{n,1,100}]
%Y A231635 Cf. A109909, A219842, A219864, A220272, A231201, A231516, A231555, A231561, A231633.
%K A231635 nonn
%O A231635 1,5
%A A231635 _Zhi-Wei Sun_, Nov 12 2013