A321611 - OEIS

%I #32 Aug 19 2021 02:01:05

%S 1,1,1,4,4,4,4,64,16,144,144,0,144,144,289,4356,2916,22500,79524,

%T 1887876,313600,3459600,2985984,50069776,32353344,2056803904,

%U 237591396,11713732900,10265337124,342040164964,30744816964,2507750953744,378640854244,53517915572836,7415600385600,230030730231696

%N Number of permutations tau of {1,...,n} such that k^4 + tau(k)^4 is prime for every k = 1,...,n.

%C Conjecture: Each term of the sequence is a positive square.

%C See also A321597 and A321610 for similar conjectures.

%C This conjecture fails for n = 12. The values of a(12),...,a(20) were first computed by the user MTson on Mathoverflow. Ilya Bogdanov has confirmed that a(n) is indeed a square. See answers and comments to Question 315351 on Mathoverflow. - _Zhi-Wei Sun_, Nov 17 2018

%H Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/315341">Primes arising from permutations (II)</a>, Question 315341 on Mathoverflow, Nov. 14, 2018.

%H Zhi-Wei Sun, <a href="https://mathoverflow.net/questions/315351">A mysterious connection between primes and squares</a>, Question 315351 on Mathoverflow, Nov. 15, 2018.

%e a(3) = 1, and (1,3,2) is a permutation of {1,...,n} with 1^4 + 1^4 = 2, 2^4 + 3^4 = 97 and 3^4 + 2^4 = 97 all prime.

%t V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]

%t Do[r=0;Do[Do[If[PrimeQ[i^4+Part[V[n],k][[i]]^4]==False,Goto[aa]],{i,1,n}];r=r+1;Label[aa],{k,1,n!}];Print[n," ",r],{n,1,11}]

%o (PARI) a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i^4 + j^4))); \\ _Jinyuan Wang_, Jun 13 2020

%Y Cf. A000040, A000583, A321597, A321610.

%K nonn,more

%O 1,4

%A _Zhi-Wei Sun_, Nov 14 2018

%E a(12)-a(27) from _Jud McCranie_, Nov 17 2018

%E a(28)-a(29) from _Jinyuan Wang_, Jun 13 2020

%E a(30)-a(36) from _Vaclav Kotesovec_, Aug 19 2021