A390695 Number of permutations f of {1,...,n} with f(1) = 1 such that the n numbers prime(k) + prime(f(k)) + 1 (k = 1..n) are distinct primes.

1, 1, 1, 0, 2, 2, 0, 4, 20, 88, 18, 0, 224, 0, 0, 1234, 60428, 5742, 0, 9276

Offset

1,5

Comments

If f is a permutation of {1,...,n} with f(1) = 1, and the n numbers prime(k) + prime(f(k)) + 1 (k = 1..n) are distinct primes, then f^{-1} (the inverse of f) is also a permutaion of {1,...,n} with f^{-1}(1) = 1, and the n numbers prime(k) + prime(f^{-1}(k)) + 1 (k = 1..n) are distinct primes.

Links

Table of n, a(n) for n=1..20.

Zhi-Wei Sun, On permutations of {1,...,n} and related topics, J. Algebraic Combin. 54 (2021), 893-912.

Example

a(5) = 2 since we can only take (f(1),...,f(5)) = (1,2,4,5,3), (1,2,5,3,4).
a(6) = 2 since we can only take (f(1),...,f(6)) = (1,4,6,3,5,2), (1,6,4,2,5,3).
a(8) = 4 since we can only take (f(1),...,f(8)) = (1,4,6,3,7,2,8,5), (1,4,6,3,8,2,5,7), (1,6,4,2,7,3,8,5), (1,6,4,2,8,3,5,7).
a(11) > 0 since we may take (f(1),...,f(11)) = (1,4,6,3,8,2,9,7,10,11,5). Note that prime(1) + prime(1) + 1 = 5, prime(2) + prime(4) + 1 = 11, prime(3) + prime(6) + 1 =19, prime(4) + prime(3) + 1 = 13, prime(5) + prime(8) + 1 = 31, prime(6) + prime(2) + 1 = 17, prime(7) + prime(9) + 1 = 41, prime(8) + prime(7) + 1 = 37, prime(9) + prime(10) + 1 = 53, prime(10) + prime(11) + 1 = 61 and prime(11) + prime(5) + 1 = 43.

Mathematica

A program to compute a(8):
p[n_]:=p[n]=Prime[n]; V[i_]:=V[i]=Part[Permutations[{2, 3, 4, 5, 6, 7, 8}], i];
m=0; Do[U={}; Do[q=p[j]+p[V[i][[j-1]]]+1; If[PrimeQ[q], U=Append[U, q]], {j, 2, 8}]; If[Length[Union[U]]==7, m=m+1], {i, 1, 7!}]; Print[m]

Crossrefs

Cf. A000040, A321727.

Sequence in context: A344913 A052079 A387477 * A387763 A291483 A181295

Adjacent sequences: A390692 A390693 A390694 * A390696 A390697 A390698

Keyword

nonn,more

Author

Zhi-Wei Sun, Nov 15 2025

Extensions

a(11)-a(20) from Alois P. Heinz, Nov 15 2025

Status

approved