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

2,4

Comments

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

Links

Table of n, a(n) for n=2..13.

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.

Example

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.

Mathematica

(* 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]

Prog

(PARI) 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

Crossrefs

Cf. A000290, A342966.

Sequence in context: A117965 A111646 A363629 * A117753 A145883 A062820

Adjacent sequences: A342962 A342963 A342964 * A342966 A342967 A342968

Keyword

nonn,more

Author

Zhi-Wei Sun, Mar 31 2021

Extensions

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

Status

approved