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A046073
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Number of squares in multiplicative group modulo n.
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6
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1, 1, 1, 1, 2, 1, 3, 1, 3, 2, 5, 1, 6, 3, 2, 2, 8, 3, 9, 2, 3, 5, 11, 1, 10, 6, 9, 3, 14, 2, 15, 4, 5, 8, 6, 3, 18, 9, 6, 2, 20, 3, 21, 5, 6, 11, 23, 2, 21, 10, 8, 6, 26, 9, 10, 3, 9, 14, 29, 2, 30, 15, 9, 8, 12, 5, 33, 8, 11, 6, 35, 3, 36, 18, 10, 9, 15, 6, 39, 4, 27, 20, 41, 3, 16, 21
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Contribution from Artur Jasinski (grafix(AT)csl.pl), Jul 03 2010: (Start)
a(n) = Number of different diagonal elements in Cayley Table for Galois Field GF(n).
That the same number of different elements are on the diagonal of the Cayley table does not mean in every case that GF are isomorphic. (End)
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REFERENCES
| Shanks, D., Solved and Unsolved Problems in Number Theory, 4th ed. New York: Chelsea, p. 95, 1993.
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LINKS
| S. R. Finch and Pascal Sebah, Square and Cubes Modulo n (arXiv:math.NT/0604465).
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.
Eric Weisstein's World of Mathematics, Quadratic Residue
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FORMULA
| A046073(n) * A060594(n) = A000010(n) = phi(n) (This gives a formula for A046073(n) using the one in A060594(n) ). - Sharon Sela (sharonsela(AT)hotmail.com), Mar 09 2002
Multiplicative with a(2^e) = 2^max(e-3,0), a(p^e) = (p-1)/2*p^(e-1) for p an odd prime.
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MATHEMATICA
| cc = {1}; Do[bb = {}; ct = {}; v = {}; Do[If[GCD[p, gf] == 1, AppendTo[v, p]], {p, 1, gf - 1}]; len = Length[v]; Do[aa = {}; Do[AppendTo[aa, Mod[v[[n]] v[[m]], gf]], {m, 1, len}]; AppendTo[ct, aa], {n, 1, len}]; Do[AppendTo[bb, ct[[n]][[n]]], {n, 1, len}]; bb = Union[bb]; AppendTo[cc, Length[bb]], {gf, 2, 100}]; cc (*Artur Jasinski*) [From Artur Jasinski (grafix(AT)csl.pl), Jul 03 2010]
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CROSSREFS
| Cf. A046072, A007735, A060594, A000010, A087692, A000224.
Sequence in context: A075825 A007735 A002616 * A162912 A039776 A048864
Adjacent sequences: A046070 A046071 A046072 * A046074 A046075 A046076
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KEYWORD
| nonn,easy,mult
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AUTHOR
| Eric Weisstein (eric(AT)weisstein.com)
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EXTENSIONS
| Edited and verified by Franklin T. Adams-Watters (FrankTAW(AT)Netscape.net) Nov 07 2006
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