A321610 Number of permutations tau of {1,...,n} such that k^2 + tau(k)^2 is prime for every k = 1,...,n.

1, 1, 1, 1, 1, 4, 0, 16, 4, 144, 64, 81, 256, 5184, 1600, 25600, 8100, 183184, 108900, 5924356, 342225, 9066121, 11356900, 106853569, 105698961, 16119349444, 1419933124, 69792129124, 14251584400, 613950602500, 304388337796, 25042678198756, 10080904401936, 1179245283899881, 1045903153861476, 31082438574307129

Offset: 1

Author

Zhi-Wei Sun, Nov 14 2018

Keywords

• nonn
• more

Comments

Conjecture 1: The number a(n) is always a square, and a(n) = 0 only for n = 7.

Conjecture 2: For any positive integer n, there is a permutation tau of {1,...,n} such that k^2 + k*tau(k) + tau(k)^2 is prime for every k = 1,...,n.

See also A321597 and A321611 for similar conjectures.

Examples

			a(3) = 1, and (1,3,2) is a permutation of {1,2,3} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13 and 3^2 + 2^2 all prime.
a(5) = 1, and (1,3,2,5,4) is a permutation of {1,2,3,4,5} with 1^2 + 1^2 = 2, 2^2 + 3^2 = 13, 3^2 + 2^2 = 13, 4^2 + 5^2 = 41 and 5^2 + 4^2 = 41 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, A000290, A321597, A321611.

Programs

Mathematica

V[n_]:=V[n]=Permutations[Table[i,{i,1,n}]]
Do[r=0;Do[Do[If[PrimeQ[i^2+Part[V[n],k][[i]]^2]==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^2 + j^2))); \\ Jinyuan Wang, Jun 13 2020

Extensions

a(12)-a(25) from Jud McCranie, Nov 15 2018

a(26)-a(28) from Jud McCranie, Nov 19 2018

a(29)-a(30) from Jinyuan Wang, Jun 13 2020

a(31)-a(36) from Vaclav Kotesovec, Aug 19 2021