A367926 a(n) is the number of 2 X 2 matrices over the integers mod n that are invertible mod n for every permutation of their elements.

4, 40, 64, 400, 160, 1704, 1024, 3240, 1600, 11560, 2560, 23280, 6816, 16000, 16384, 71104, 12960, 112680, 25600, 68160, 46240, 247720, 40960, 250000, 93120, 262440, 109056, 641200, 64000, 842280, 262144, 462400, 284416, 681600, 207360, 1733904, 450720, 931200, 409600, 2633440, 272640, 3196200, 739840

Offset

2,1

Comments

a(n) is the number of 4-tuples [a,b,c,d] of integers in [0,1,...,n-1] such that a*b-c*d, a*c-b*d and a*d-b*c are all coprime to n.

Links

Robert Israel, Table of n, a(n) for n = 2..10000

MathOverflow, Matrices over F_p that have nonzero determinant under any element permutation.

Formula

Multiplicative with a(p^k) = p^(4*(k-1)) * (p^4 - 3*p^3 + 9*p^2 - 17*p + 10) for primes p.

Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = Product_{p prime} (1 - 3/p^2 + 9/p^3 - 17/p^4 + 10/p^5) = 0.42729799106430918317... . - Amiram Eldar, Jan 20 2024

Example

a(3) = 40 because the 4-tuples [0, 1, 1, 1], [0, 1, 1, 2], [1, 1, 1, 2], [0, 1, 2, 2], [0, 2, 2, 2], [1, 2, 2, 2] and their permutations satisfy the criterion for n = 4. Of these, [0,1,1,2] and [0,1,2,2] each have 12 permutations and the others have 4, so a(3) = 2 * 12 + 4 * 4 = 40.

Maple

filter:= proc(L, n)
 andmap(t -> igcd(t, n)=1, {L[1]*L[2]-L[3]*L[4], L[1]*L[3]-L[2]*L[4], L[1]*L[4]-L[2]*L[3]})
end proc:
g:= proc(n) local a, b, c, d, t; option remember;
  t:= 0;
  for a from 0 to n-1 do
    for b from 0 to a do
      for c from 0 to b do
        for d from 0 to c do
          if filter([a, b, c, d], n) then t:= t + combinat:-numbperm([a, b, c, d]) fi
  od od od od;
  t
end proc:
f:= proc(n) local F, t;
  F:= ifactors(n)[2];
  mul(g(t[1]^t[2]), t=F)
end proc:
map(f, [$2..50]);

Mathematica

f[p_, e_] := p^(4*(e - 1))*(p^4 - 3*p^3 + 9*p^2 - 17*p + 10); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100, 2] (* Amiram Eldar, Jan 20 2024 *)

Prog

(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, p=f[i, 1]; e=f[i, 2]; p^(4*(e-1)) * (p^4 - 3*p^3 + 9*p^2 - 17*p + 10)); } \\ Amiram Eldar, Jan 20 2024

Crossrefs

Sequence in context: A077329 A169967 A061473 * A091104 A069539 A063997

Adjacent sequences: A367923 A367924 A367925 * A367927 A367928 A367929

Keyword

nonn,mult

Author

Robert Israel, Dec 05 2023

Status

approved