A346387 Number of permutations f of {1,...,prime(n)-1} with f(prime(n)-1) = prime(n)-1 and f(prime(n)-2) = prime(n)-2 such that 1/(f(1)*f(2)) + 1/(f(2)*f(3)) + ... + 1/(f(prime(n)-2)*f(prime(n)-1)) + 1/(f(prime(n)-1)*f(1)) == 0 (mod prime(n)^2).
0, 1, 1, 323, 21615, 301654585
Offset: 2
Examples
a(3) = 1, and 1/(2*1) + 1/(1*3) + 1/(3*4) + 1/(4*2) = 5^2/24 == 0 (mod 5^2). a(4) = 1, and 1/(2*3) + 1/(3*4) + 1/(4*1) + 1/(1*5) + 1/(5*6) + 1/(6*2) = 7^2/60 == 0 (mod 7^2). a(5) > 0, and 1/(1*2) + 1/(2*4) + 1/(4*6) + 1/(6*3) + 1/(3*5) + 1/(5*7) + 1/(7*8) + 1/(8*9) + 1/(9*10) + 1/(10*1) = 11^2/126 == 0 (mod 11^2). a(6) > 0, and 1/(1*2) + 1/(2*3) + 1/(3*7) + 1/(7*4) + 1/(4*9) + 1/(9*5) + 1/(5*8) + 1/(8*10) + 1/(10*6) + 1/(6*11) + 1/(11*12) + 1/(12*1) = 13^2/176 ==0 (mod 13^2).
Programs
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Mathematica
(* A program to compute a(5): *) VV[i_]:=Part[Permutations[{1,2,3,4,5,6,7,8}],i]; rMod[m_,n_]:=Mod[Numerator[m]*PowerMod[Denominator[m],-1,n],n,-n/2]; n=0;Do[If[rMod[Sum[1/(VV[i][[k]]VV[i][[k+1]]),{k,1,7}]+1/(VV[i][[8]]*9)+1/(9*10)+1/(10*VV[i][[1]]),11^2]==0,n=n+1],{i,1,8!}];Print[n] a[n_] := Block[{p = Prime@n, inv}, inv = ModularInverse[#, p^2] & /@ Range[p-1]; Length@ Select[ Join[#, Take[inv, -2]] & /@ Permutations[ Take[inv, p-3]], Mod[#[[1]] #[[-1]] + Total[Times @@@ Partition[#, 2, 1]], p^2] == 0 &]]; a /@ Range[2, 6] (* Giovanni Resta, Jul 15 2021 *)
Extensions
a(6)-a(7) from Giovanni Resta, Jul 15 2021
Comments