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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, 0, 1, 3, 2, 27, 44, 154, 1687, 2925
OFFSET
2,6
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.
LINKS
EXAMPLE
a(6) = 1, and 1^1 * 2^2 * 3^5 * 4^6 * 5^3 * 6^4 = (8461-1)^3 with 8461 prime.
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]
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Zhi-Wei Sun, Apr 05 2021
EXTENSIONS
a(12)-a(13) from David A. Corneth, Apr 06 2021
STATUS
approved