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Search: id:A000086
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| A000086 |
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Number of solutions to x^2 - x + 1 == 0 (mod n). |
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+0
10
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| 1, 0, 1, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 4, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 2, 0, 0
(list; graph; listen)
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OFFSET
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1,7
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COMMENT
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Number of elliptic points of order 3 for GAMMA_0 (n).
Equivalently, number of fixed points of GAMMA_0 (n) of type rho.
Values are 0 or a power of 2.
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REFERENCES
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Fell, Harriet; Newman, Morris; Ordman, Edward; Tables of genera of groups of linear fractional transformations. J. Res. Nat. Bur. Standards Sect. B 67B 1963 61-68.
B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3).
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LINKS
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Christian G. Bower, Table of n, a(n) for n=1..2000
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FORMULA
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Multiplicative with a(p^e) = 1 if p = 3 and e = 1; 0 if p = 3 and e > 1; 2 if p == 1 (mod 3); 0 if p == 2 (mod 3). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
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MAPLE
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with(numtheory); A000086 := proc (n) local d, s; if modp(n, 9) = 0 then RETURN(0) fi; s := 1; for d in divisors(n) do if isprime(d) then s := s*(1+eval(legendre(-3, d))) fi od; s end: (Gene Smith, May 22 2006)
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MATHEMATICA
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Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
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PROGRAM
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(PARI) a(n)=if(n<1, 0, sum(x=0, n-1, (x^2-x+1)%n==0))
(PARI) a(n)=if(n<1, 0, direuler(p=2, n, if(p==3, 1+X, if(p%3==2, 1, (1+X)/(1-X))))[n])
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CROSSREFS
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Cf. A000089, A000091, A001616, A014683.
Sequence in context: A030201 A055668 A045839 this_sequence A045838 A045837 A126825
Adjacent sequences: A000083 A000084 A000085 this_sequence A000087 A000088 A000089
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KEYWORD
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nonn,easy,nice,mult
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AUTHOR
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njas
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