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

%I A066907 #19 Nov 05 2023 09:03:16
%S A066907 1,4,14,28,32,56,58,176,110,128,134,392,184,232,448,608,308,440,382,
%T A066907 896,812,536,554,2464,752,736,974,1624,872,1792,994,2336,1876,1232,
%U A066907 1856,3080,1408,1528,2576,5632,1724,3248,1894,3752,3520,2216,2258,8512,2746
%N A066907 Number of elements in GL(2,Z_n) x with x^2 == I mod n where I is the identity matrix.
%C A066907 Number of involutory matrices mod n. - _Charles R Greathouse IV_, May 29 2013
%H A066907 Charles R Greathouse IV, <a href="/A066907/b066907.txt">Table of n, a(n) for n = 1..10000</a>
%F A066907 a(n) = A066947(n) + 1.
%F A066907 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]
%F A066907 From _Amiram Eldar_, Nov 03 2023: (Start)
%F A066907 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).
%F A066907 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)
%t A066907 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 *)
%o A066907 (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
%Y A066907 Cf. A066947, A060594, A002117.
%K A066907 nonn,mult,easy
%O A066907 1,2
%A A066907 Sharon Sela (sharonsela(AT)hotmail.com), Jan 26 2002
%E A066907 Added more terms (from A066947), _Joerg Arndt_, Mar 08 2013