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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. 5
 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 (list; graph; refs; listen; history; text; internal format)
 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

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Last modified June 19 19:59 EDT 2024. Contains 373507 sequences. (Running on oeis4.)