A321727 Number of permutations f of {1,...,n} such that prime(k) + prime(f(k)) + 1 is prime for every k = 1,...,n.

1, 1, 1, 2, 6, 10, 31, 76, 696, 4294, 5772, 8472, 128064, 147960, 1684788, 26114739, 523452320, 1029877159, 1772807946, 28736761941, 19795838613, 31445106424, 1313504660737, 54477761675626, 105122845176663, 2200119900732333, 2761739099984389, 83123428119278837, 219734505495953342, 7228968492870136475, 13623311188546432233, 625620139149376861330, 18603738861035365389401, 64952397216275572992159, 3115094155636931821691880, 4788927142804364353625983

Offset: 1

Author

Zhi-Wei Sun, Nov 17 2018

Keywords

• nonn

Comments

Clearly, a(n) is also the permanent of the matrix of order n whose (i,j)-entry is 1 or 0 according as prime(i) + prime(j) + 1 is prime or not.

Conjecture: a(n) > 0 for all n > 0.

Note that there is no permutation f of {1,...,10} such that prime(k) + prime(f(k)) - 1 is prime for every k = 1,...,10.

Examples

			a(3) = 1, and (1,2,3) is a permutation of {1,2,3} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(2) + 1 = 7 and prime(3) + prime(3) + 1 = 11 all prime.
a(4) = 2. In fact, (1,2,4,3) is a permutation of {1,2,3,4} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(2) + 1 = 7, prime(3) + prime(4) + 1 = 13 and prime(4) + prime(3) + 1 = 13 all prime; also (1,4,3,2) is a permutation of {1,2,3,4} with prime(1) + prime(1) + 1 = 5, prime(2) + prime(4) + 1 = 11, prime(3) + prime(3) + 1 = 11 and prime(4) + prime(2) + 1 = 11 all prime.
		

Links

• Zhi-Wei Sun, Permutations pi with p_k+p_{pi(k)}+1 prime for all k = 1,...,n, Question 315581 on Mathoverflow, Nov. 17, 2018.

Crossrefs

Cf. A000040, A321597, A321610, A321611.

Programs

Maple

b:= proc(s) option remember; (k-> `if`(k=0, 1, add(`if`(isprime(
      ithprime(i)+ithprime(k)+1), b(s minus {i}), 0), i=s)))(nops(s))
    end:
a:= n-> b({$1..n}):
seq(a(n), n=1..15);  # Alois P. Heinz, Nov 17 2018

Mathematica

p[n_]:=p[n]=Prime[n];
a[n_]:=a[n]=Permanent[Table[Boole[PrimeQ[p[i]+p[j]+1]],{i,1,n},{j,1,n}]];
Do[Print[n," ",a[n]],{n,1,22}]

PARI

a(n) = matpermanent(matrix(n, n, i, j, ispseudoprime(prime(i)+prime(j)+1))); \\ Jinyuan Wang, Jun 13 2020

Extensions

a(23)-a(26) from Alois P. Heinz, Nov 17 2018

a(27)-a(28) from Jinyuan Wang, Jun 13 2020

a(29)-a(36) from Vaclav Kotesovec, Aug 19 2021