Search: id:A053191
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%I A053191
%S A053191 1,4,18,32,100,72,294,256,486,400,1210,576,2028,1176,1800,2048,4624,
%T A053191 1944,6498,3200,5292,4840,11638,4608,12500,8112,13122,9408,23548,7200,
%U A053191 28830,16384,21780,18496,29400,15552,49284,25992,36504,25600,67240
%N A053191 a(n)=n^2*phi(n).
%C A053191 Number of invertible 2 X 2 symmetric matrices over Z(n). - T. D. Noe 
               (noe(AT)sspectra.com), Jan 13 2006
%C A053191 Note that A115077 gives the number of 2 X 2 symmetric matrices having 
               nonzero determinant. However for composite n a nonzero determinant 
               is not sufficient for the matrix to be invertible; the determinant 
               must also be relatively prime to n. - T. D. Noe (noe(AT)sspectra.com), 
               Jan 13 2006
%C A053191 Also Euler phi function of n^3.
%C A053191 For n^k, EulerPhi[n^k]=n^(k-1)*EulerPhi[n]. The same holds if Phi is 
               replaced by cototient function.
%C A053191 Also the sum of the degrees of the irreducible representations of the 
               group GL(2,Z_n) (sequence A000252). - Sharon Sela (sharonsela(AT)hotmail.com), 
               Feb 06 2002
%F A053191 a(n)=n^2*EulerPhi[n]=A000010(n^3)
%e A053191 n=5: n^3=125, EulerPhi[125]=125-25=100.
%p A053191 with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); - Zerinvary Lajos 
               (zerinvarylajos(AT)yahoo.com), Oct 07 2007
%t A053191 Table[cnt=0; Do[m={{a, b}, {b, c}}; If[Det[m, Modulus->n]>0 && MatrixQ[Inverse[m, 
               Modulus->n]], cnt++ ], {a, 0, n-1}, {b, 0, n-1}, {c, 0, n-1}]; cnt, 
               {n, 2, 50}] - T. D. Noe (noe(AT)sspectra.com), Jan 13 2006
%t A053191 Table[EulerPhi[n^3],{n,0,40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), 
               Nov 10 2009]
%o A053191 (Other) sage: [euler_phi(n^3)for n in xrange(1,42)]# [From Zerinvary 
               Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
%Y A053191 Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, 
               A115076, A115077.
%Y A053191 Cf. A000010, A051953, A002618, A053650, A053191, A053192, A001248.
%Y A053191 Sequence in context: A033166 A049726 A130656 this_sequence A003474 A095823 
               A092116
%Y A053191 Adjacent sequences: A053188 A053189 A053190 this_sequence A053192 A053193 
               A053194
%K A053191 nonn,mult
%O A053191 1,2
%A A053191 Labos E. (labos(AT)ana.sote.hu), Mar 02 2000
%E A053191 Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion 
               of Andrew Plewe, Jun 05 2007

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