A343130 Number of permutations tau of {1,...,n} with tau(1) = 1 and tau(2) = 2 such that Product_{k=1..n} k^tau(k) = (p-1)^3 for some prime p.

0, 0, 0, 0, 1, 3, 2, 27, 44, 154, 1687, 2925

Offset: 2

Author

Zhi-Wei Sun, Apr 05 2021

Keywords

• nonn
• more

Comments

Conjecture 1: a(n) > 0 for all n > 5.

Conjecture 2: For any integer n > 5, there is a permutation tau of {1,...,n} with tau(1) = n - 1 and tau(n) = n such that tau(1)^tau(2)*...*tau(n-1)^tau(n)*tau(n)^tau(1) = q^2 for some integer q with q - 1 and q + 1 twin prime.

Examples

			a(6) = 1, and 1^1 * 2^2 * 3^5 * 4^6 * 5^3 * 6^4 = (8461-1)^3 with 8461 prime.
		

Links

• Zhi-Wei Sun, On permutations of {1,...,n} and related topics, J. Algebraic Combin., 2021.

Crossrefs

Cf. A000040, A000290, A000578, A014574, A342965, A342966.

Programs

Mathematica

(* A program to compute a(8): *)
CQ[n_]:=CQ[n]=IntegerQ[n^(1/3)]&&PrimeQ[n^(1/3)+1]
V[i_]:=V[i]=Part[Permutations[{3,4,5,6,7,8}], i]
S[i_]:=S[i]=4*Product[(j+2)^(V[i][[j]]),{j,1,6}]
n=0;Do[If[CQ[S[i]],n=n+1],{i,1,6!}];Print[8," ",n]

Extensions

a(12)-a(13) from David A. Corneth, Apr 06 2021