A066907 Number of elements in GL(2,Z_n) x with x^2 == I mod n where I is the identity matrix.

1, 4, 14, 28, 32, 56, 58, 176, 110, 128, 134, 392, 184, 232, 448, 608, 308, 440, 382, 896, 812, 536, 554, 2464, 752, 736, 974, 1624, 872, 1792, 994, 2336, 1876, 1232, 1856, 3080, 1408, 1528, 2576, 5632, 1724, 3248, 1894, 3752, 3520, 2216, 2258, 8512, 2746

Offset

1,2

Comments

Number of involutory matrices mod n. - Charles R Greathouse IV, May 29 2013

Links

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

Formula

a(n) = A066947(n) + 1.

a(n) is multiplicative and for an odd prime power p^k : a(p^k) = 2 + p^(2k-1)(p+1). [corrected by Felix A. Pahl, Mar 08 2013]

From Amiram Eldar, Nov 03 2023: (Start)

Dirichlet g.f.: ((1+1/2^s+7/2^(2*s-1)+5/2^(3*s-4))/(1+5/2^s)) * (zeta(s)*zeta(s-2)/zeta(s-1)) * Product_{p prime} (1 + 2/p^(s-1) + 1/p^s).

Sum_{k=1..n} a(k) ~ c * n^3, where c = (4*zeta(3)/13) * Product_{p prime} (1 + 1/p^2 + 1/p^3 - 2/p^4 - 1/p^5) = 0.55646002711570137209... . (End)

Mathematica

f[p_, e_] := p^(2*e-1)*(p+1) + 2; f[2, e_] := 9*4^(e-1)+32; f[2, 1] = 4; f[2, 2] = 28; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 02 2023 *)

Prog

(PARI) a(n)=my(o=valuation(n, 2), f=factor(n>>o)); prod(i=1, #f[, 1], f[i, 1]^(2*f[i, 2])+f[i, 1]^(2*f[i, 2]-1)+2)*if(o, if(o>1, if(o>2, 9*4^(o-1)+32, 28), 4), 1) \\ Charles R Greathouse IV, May 29 2013

Crossrefs

Cf. A066947, A060594, A002117.

Sequence in context: A094320 A200546 A304342 * A130439 A289179 A033690

Adjacent sequences: A066904 A066905 A066906 * A066908 A066909 A066910

Keyword

nonn,mult,easy

Author

Sharon Sela (sharonsela(AT)hotmail.com), Jan 26 2002

Extensions

Added more terms (from A066947), Joerg Arndt, Mar 08 2013

Status

approved