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 A000086 Number of solutions to x^2 - x + 1 == 0 (mod n). 29
 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; refs; listen; history; text; internal format)
 OFFSET 1,7 COMMENTS 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. Shadow transform of central polygonal numbers A002061. - Michel Marcus, Jun 06 2013 a(A226946(n)) = 0; a(A034017(n)) > 0. - Reinhard Zumkeller, Jun 23 2013 Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - John M. Campbell, Apr 01 2018 From Jianing Song, Jul 03 2018: (Start) The comment above is true. Since both a(n) and A001615(n) are multiplicative we just have to verify that for prime powers. Note that A001615(p^e) = (p+1)*p^(e-1). For p == 1 (mod 3), p+1 == 2 (mod 3) so (p+1)*p^(e-1) == 2 (mod 3); for p == 2 (mod 3), p+1 is a multiple of 3 so (p+1)*p^(e-1) == 0 (mod 3). For p = 3, if e = 1 then p+1 == 1 (mod 3); if e > 1 then (p+1)*p^(e-1) == 0 (mod 3). Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End) REFERENCES G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3). B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101. LINKS Christian G. Bower, Table of n, a(n) for n = 1..2000 Harriet Fell, Morris Newman, and Edward Ordman, Tables of genera of groups of linear fractional transformations, J. Res. Nat. Bur. Standards Sect. B 67B (1963), 61-68. Lorenz Halbeisen and Norbert Hungerbuehler, Number theoretic aspects of a combinatorial function, Notes on Number Theory and Discrete Mathematics 5(4) (1999), 138-150; see Definition 7 for the shadow transform. John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. A65 (2009), 156-163. [See Table 4.] N. J. A. Sloane, Transforms. FORMULA 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, Aug 01 2001 a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - Michael Somos, Aug 14 2015 EXAMPLE G.f. = x + x^3 + 2*x^7 + 2*x^13 + 2*x^19 + 2*x^21 + 2*x^31 + 2*x^37 + 2*x^39 + ... MAPLE 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 Ward Smith, May 22 2006 MATHEMATICA Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ] a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* Michael Somos, Aug 14 2015 *) a[n_] := a[n] = Product[{p, e} = pe; Which[p==1 || p==3 && e==1, 1, p==3 && e>1, 0, Mod[p, 3]==1, 2, Mod[p, 3]==2, 0, True, a[p^e]], {pe, FactorInteger[n]}]; Array[a, 105] (* Jean-François Alcover, Oct 18 2018 *) PROG (PARI) {a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; /* Nov 15 2002 */ (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])}; /* Nov 15 2002 */ (Haskell) a000086 n = if n `mod` 9 == 0 then 0   else product \$ map ((* 2) . a079978 . (+ 2)) \$ a027748_row \$ a038502 n -- Reinhard Zumkeller, Jun 23 2013 CROSSREFS Cf. A000089, A000091, A001616, A014683. Cf. A027748, A079978, A038502, A007949. Cf. A341422 (without zeros). Sequence in context: A030201 A055668 A045839 * A045838 A293814 A045837 Adjacent sequences:  A000083 A000084 A000085 * A000087 A000088 A000089 KEYWORD nonn,easy,nice,mult,changed AUTHOR STATUS approved

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Last modified April 22 22:19 EDT 2021. Contains 343197 sequences. (Running on oeis4.)