A115075 - OEIS

%I #10 Oct 31 2023 11:14:53

%S 1,4,9,20,25,36,49,80,99,100,121,180,169,196,225,352,289,396,361,500,

%T 441,484,529,720,725,676,891,980,841,900,961,1408,1089,1156,1225,1980,

%U 1369,1444,1521,2000,1681,1764,1849,2420,2475,2116,2209,3168,2695,2900

%N Number of 2 X 2 symmetric matrices over Z(n) having determinant 0.

%F a(n) = n^3 - A115077(n).

%F For squarefree n, a(n) = n^2.

%F Multiplicative with a(p^e) = p^(e)*(p^(e)+p^(e-1)-p^(ceiling(e/2)-1)).

%F From _Amiram Eldar_, Oct 31 2023: (Start)

%F Dirichlet g.f.: zeta(s-2) * zeta(2*s-3) / zeta(2*s-2).

%F Sum_{k=1..n} a(k) ~ (zeta(3)/(3*zeta(4))) * n^3. (End)

%t Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]==0, cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, {n, 50}]

%t f[p_, e_] := p^e*(p^e + p^(e-1) - p^(Ceiling[e/2] - 1)); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]  (* _Amiram Eldar_, Oct 31 2023 *)

%o (PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; p^e*(p^e + p^(e-1) - p^((e+1)\2 - 1)));} \\ _Amiram Eldar_, Oct 31 2023

%Y Cf. A020478 (number of singular 2 X 2 matrices over Z(n)), A115077.

%Y Cf. A002117, A013662.

%K mult,nonn,easy

%O 1,2

%A _T. D. Noe_, Jan 12 2006