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%I #14 Aug 07 2023 02:30:36
%S 0,0,0,1,1,2,0,2,3,2,1,2,1,1,4,4,1,2,2,3,3,2,2,5,1,4,1,1,5,4,1,2,5,5,
%T 3,8,3,6,5,5,4,4,2,4,5,3,1,8,3,4,4,1,2,8,6,3,4,5,4,4,7,1,3,6,5,7,3,3,
%U 8,2,4,5,2,6,10,7,1,5,5,6,8,6,4,5,5,7,5,4,4,11,4,5,5,5,6,6,3,1,12,8
%N Number of ways to write n=x+y (x>0, y>0) with 3x-2, 3x+2 and 2xy+1 all prime
%C Conjecture: a(n)>0 for all n>7.
%C This has been verified for n up to 10^8. It implies that there are infinitely many cousin primes.
%C Conjecture verified for n up to 10^9. - _Mauro Fiorentini_, Aug 06 2023
%C Zhi-Wei Sun also made some other similar conjectures, e.g., he conjectured that any integer n>17 can be written as x+y (x>0, y>0) with 2x-3, 2x+3 and 2xy+1 all prime, and each integer n>28 can be written as x+y (x>0, y>0) with 2x+1, 2y-1 and 2xy+1 all prime.
%C Both conjectures verified for n up to 10^9. - _Mauro Fiorentini_, Aug 06 2023
%H Zhi-Wei Sun, <a href="/A220455/b220455.txt">Table of n, a(n) for n = 1..10000</a>
%H Zhi-Wei Sun, <a href="http://arxiv.org/abs/1211.1588">Conjectures involving primes and quadratic forms</a>, arXiv:1211.1588 [math.NT], 2012-2017.
%e a(25)=1 since 25=13+12 with 3*13-2, 3*13+2 and 2*13*12+1=313 all prime.
%t a[n_]:=a[n]=Sum[If[PrimeQ[3k-2]==True&&PrimeQ[3k+2]==True&&PrimeQ[2k(n-k)+1]==True,1,0],{k,1,n-1}]
%t Do[Print[n," ",a[n]],{n,1,1000}]
%t apQ[{a_,b_}]:=AllTrue[{3a-2,3a+2,2a*b+1},PrimeQ]; Table[Count[Flatten[ Permutations/@ IntegerPartitions[n,{2}],1],_?(apQ[#]&)],{n,100}] (* The program uses the AllTrue function from Mathematica version 10 *) (* _Harvey P. Dale_, Jun 09 2018 *)
%Y Cf. A220431, A023200, A046132, A218867, A219055, A220419, A220413, A220272, A219842, A219864, A219923.
%K nonn
%O 1,6
%A _Zhi-Wei Sun_, Dec 15 2012
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