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A000089
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Number of solutions to x^2 + 1 == 0 (mod n).
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9
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1, 1, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0
(list; graph; refs; listen; history; internal format)
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OFFSET
| 1,5
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COMMENTS
| Number of elliptic points of order 2 for GAMMA_0 (n).
The Dirichlet inverse, 1, -1, 0, 1, -2, 0, 0, -1, 0, 2, 0, 0, -2, 0,.. seems to equal A091400, apart from signs. [From R. J. Mathar (mathar(AT)strw.leidenuniv.nl), Jul 15 2010]
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REFERENCES
| M. Baake, "Solution of coincidence problem...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.
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.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156163. [See Table 4].
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LINKS
| T. D. Noe, Table of n, a(n) for n=1..2000
M. Baake and U. Grimm, Quasicrystalline combinatorics
S. R. Finch and Pascal Sebah, Squares and Cubes Modulo n (arXiv:math.NT/0604465).
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FORMULA
| a(n) = 0 if 4|n, else a(n) = Product_{ p | N } (1 + Legendre(-1, p) ), where we use the definition that Legendre(-1, 2) = 0, Legendre(-1, p) = 1 if p == 1 mod 4, = -1 if p == 3 mod 4. This is Shimura's definition, which is different from Maple's.
Dirichlet series: (1+2^(-s))*Product (1+p^(-s))/(1-p^(-s)) (p=1 mod 4).
Multiplicative with a(p^e) = 1 if p = 2 and e = 1; 0 if p = 2 and e > 1; 2 if p == 1 (mod 4); 0 if p == 3 (mod 4). - David W. Wilson (davidwwilson(AT)comcast.net), Aug 01, 2001.
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MAPLE
| with(numtheory); A000089 := proc (n) local i, s; if modp(n, 4) = 0 then RETURN(0) fi; s := 1; for i in divisors(n) do if isprime(i) and i > 2 then s := s*(1+eval(legendre(-1, i))) fi od; s end: (Gene Smith, May 22 2006)
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MATHEMATICA
| Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 3 ]==2, 0, Count[ Array[ Mod[ #^2+1, n ]&, n, 0 ], 0 ] ] ], 84 ]
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CROSSREFS
| Sequence in context: A001343 A022882 A171919 * A051907 A178176 A093569
Adjacent sequences: A000086 A000087 A000088 * A000090 A000091 A000092
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KEYWORD
| nonn,nice,mult
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AUTHOR
| N. J. A. Sloane (njas(AT)research.att.com).
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