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A053191
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a(n)=n^2*phi(n).
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8
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1, 4, 18, 32, 100, 72, 294, 256, 486, 400, 1210, 576, 2028, 1176, 1800, 2048, 4624, 1944, 6498, 3200, 5292, 4840, 11638, 4608, 12500, 8112, 13122, 9408, 23548, 7200, 28830, 16384, 21780, 18496, 29400, 15552, 49284, 25992, 36504, 25600, 67240
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,2
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COMMENTS
| Number of invertible 2 X 2 symmetric matrices over Z(n). - T. D. Noe (noe(AT)sspectra.com), Jan 13 2006
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
Also Euler phi function of n^3.
For n^k, EulerPhi[n^k]=n^(k-1)*EulerPhi[n]. The same holds if Phi is replaced by cototient function.
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
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FORMULA
| a(n)=n^2*EulerPhi[n]=A000010(n^3).
Dirichlet g.f. zeta(s-3)/zeta(s-2). - R. J. Mathar, Feb 09 2011
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EXAMPLE
| n=5: n^3=125, EulerPhi[125]=125-25=100.
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MAPLE
| with(numtheory):a:=n->phi(n^3): seq(a(n), n=1..41); - Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Oct 07 2007
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MATHEMATICA
| 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
Table[n^2*EulerPhi[n], {n, 0, 40}] [From Vladimir Orlovsky (4vladimir(AT)gmail.com), Nov 10 2009]
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PROG
| (Sage) [n^2*euler_phi(n) for n in xrange(1, 42)]# [From Zerinvary Lajos (zerinvarylajos(AT)yahoo.com), Jun 06 2009]
(MAGMA) [ n^2*EulerPhi(n) : n in [1..100] ]; // Vincenzo Librandi, Apr 21 2011
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CROSSREFS
| Cf. A000252 (number of invertible 2 X 2 matrices over Z(n)), A115075, A115076, A115077.
Cf. A000010, A051953, A002618, A053650, A053191, A053192, A001248.
Sequence in context: A033166 A049726 A130656 * A003474 A095823 A092116
Adjacent sequences: A053188 A053189 A053190 * A053192 A053193 A053194
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KEYWORD
| nonn,mult
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AUTHOR
| Labos E. (labos(AT)ana.sote.hu), Mar 02 2000
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EXTENSIONS
| Edited by N. J. A. Sloane (njas(AT)research.att.com) at the suggestion of Andrew Plewe, Jun 05 2007
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