A321611 Number of permutations tau of {1,...,n} such that k^4 + tau(k)^4 is prime for every k = 1,...,n.

1, 1, 1, 4, 4, 4, 4, 64, 16, 144, 144, 0, 144, 144, 289, 4356, 2916, 22500, 79524, 1887876, 313600, 3459600, 2985984, 50069776, 32353344, 2056803904, 237591396, 11713732900, 10265337124, 342040164964, 30744816964, 2507750953744, 378640854244, 53517915572836, 7415600385600, 230030730231696

Offset: 1

Author

Zhi-Wei Sun, Nov 14 2018

Keywords

• nonn
• more

Comments

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

See also A321597 and A321610 for similar conjectures.

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

Examples

			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.
		

Links

• Zhi-Wei Sun, Primes arising from permutations (II), Question 315341 on Mathoverflow, Nov. 14, 2018.
• Zhi-Wei Sun, A mysterious connection between primes and squares, Question 315351 on Mathoverflow, Nov. 15, 2018.

Crossrefs

Cf. A000040, A000583, A321597, A321610.

Programs

Mathematica

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

PARI

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(i^4 + j^4))); \\ Jinyuan Wang, Jun 13 2020

Extensions

a(12)-a(27) from Jud McCranie, Nov 17 2018

a(28)-a(29) from Jinyuan Wang, Jun 13 2020

a(30)-a(36) from Vaclav Kotesovec, Aug 19 2021