A342966 -id:A342966 - cp's OEIS frontend

A342965 Number of permutations tau of {1,...,n} with tau(n) = n such that tau(1)^tau(2) + ... + tau(n-1)^tau(n) + tau(n)^tau(1) is a square.

0, 0, 1, 2, 1, 6, 6, 10, 27, 105, 245, 525

Offset: 2

Zhi-Wei Sun, Mar 31 2021

Conjecture: a(n) > 0 for all n > 3.

			a(4) = 1 with 2^1 + 1^3 + 3^4 + 4^2 = 10^2.
a(5) = 2 with 2^4 + 4^1 + 1^3 + 3^5 + 5^2 = 17^2 and 3^4 + 4^2 + 2^1 + 1^5 + 5^3 = 15^2.
a(6) = 1 with 1^5 + 5^2 + 2^4 + 4^3 + 3^6 + 6^1 = 29^2.
a(10) > 0 since 1^8 + 8^4 + 4^9 + 9^3 + 3^7 + 7^6 + 6^5 + 5^2 + 2^10 + 10^1 = 629^2.
a(11) > 0 since 1^3 + 3^2 + 2^10 + 10^5 + 5^7 + 7^8 + 8^6 + 6^9 + 9^4 + 4^11 + 11^1 = 4526^2.
a(12) > 0 since 1^2 + 2^5 + 5^6 + 6^8 + 8^4 + 4^11 + 11^9 + 9^7 + 7^10 + 10^3 + 3^12 + 12^1 = 51494^2.

Zhi-Wei Sun, On permutations of {1,...,n} and related topics, J. Algebraic Combin., 2021.
Zhi-Wei Sun, On the equations x^y*y^z=z^x and w^x+x^y+y^z=z^w, Question 387042 at MathOverflow, March 21, 2021.

Cf. A000290, A342966.

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

a(n) = my(c=0, v); for(i=0, (n-1)!-1, v=numtoperm(n, i); if(issquare(sum(k=2, n, v[k-1]^v[k]) + v[n]^v[1]), c++)); c; \\ Jinyuan Wang, Apr 02 2021

a(11)-a(13) from Jinyuan Wang, Apr 02 2021

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.

Original entry on oeis.org

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

Offset: 2

Zhi-Wei Sun, Apr 05 2021

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.

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

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

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

(* 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]

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

cp's OEIS Frontend

A342965 Number of permutations tau of {1,...,n} with tau(n) = n such that tau(1)^tau(2) + ... + tau(n-1)^tau(n) + tau(n)^tau(1) is a square.

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