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A321597 Number of permutations tau of {1,...,n} such that k*tau(k) + 1 is prime for every k = 1,...,n.

%I A321597 #36 Aug 19 2021 01:55:58
%S A321597 1,2,1,6,1,24,9,38,36,702,196,7386,3364,69582,45369,885360,110224,
%T A321597 14335236,640000,19867008,11009124,1288115340,188485441,17909627257,
%U A321597 4553145529,363106696516,149376066064,11141446425852,990882875761,371060259505399,16516486146304,1479426535706319,497227517362801,102319410607145600,32589727661167504,12597253470226980096
%N A321597 Number of permutations tau of {1,...,n} such that k*tau(k) + 1 is prime for every k = 1,...,n.
%C A321597 Conjecture: (i) a(n) > 0 for all n > 0. Similarly, for any integer n > 2, there is a permutation tau of {1,...,n} such that k*tau(k) - 1 is prime for every k = 1,...,n.
%C A321597 (ii) For any integer n > 2, there is a permutation tau of {1,...,n} such that k + tau(k) - 1 and k + tau(k) + 1 are twin prime for every k = 1,...,n.
%C A321597 Obviously, part (ii) of this conjecture implies the twin prime conjecture. P. Bradley proved in arXiv:1809.01012 that for any positive integer n there is a permutation tau of {1,...,n} such that k + tau(k) is prime for every k = 1,...,n.
%H A321597 Paul Bradley, <a href="http://arxiv.org/abs/1809.01012">Prime number sums</a>, arXiv:1809.01012 [math.GR], 2018.
%H A321597 Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/315259">Primes arising from permutations</a>, Question 315259 on Mathoverflow, Nov. 14, 2018.
%H A321597 Zhi-Wei Sun, <a href="https://arxiv.org/abs/1811.10503">On permutations of {1, ..., n} and related topics</a>, arXiv:1811.10503 [math.CO], 2018.
%e A321597 a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1*1 + 1 = 2, 2*3 + 1 = 7 and 3*2 + 1 = 7 all prime.
%e A321597 a(5) = 1, and (1,5,4,3,2) is a permutation of {1,2,3,4,5} with 1*1 + 1 = 2, 2*5 + 1 = 11, 3*4 + 1 = 13, 4*3 + 1 = 13 and 5*2 + 1 = 11 all prime.
%t A321597 V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]
%t A321597 tab={};Do[r=0;Do[Do[If[PrimeQ[i*Part[V[n],k][[i]]+1]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];tab=Append[tab,r],{n,1,11}]
%o A321597 (PARI) a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i*j + 1))); \\ _Jinyuan Wang_, Jun 13 2020
%Y A321597 Cf. A000040, A001359, A006512, A014574.
%K A321597 nonn,more
%O A321597 1,2
%A A321597 _Zhi-Wei Sun_, Nov 14 2018
%E A321597 a(12)-a(26) from _Alois P. Heinz_, Nov 17 2018
%E A321597 a(27)-a(30) from _Jinyuan Wang_, Jun 13 2020
%E A321597 a(31)-a(36) from _Vaclav Kotesovec_, Aug 19 2021