# Greetings from The On-Line Encyclopedia of Integer Sequences! http://oeis.org/ Search: id:a000086 Showing 1-1 of 1 %I A000086 #81 Oct 11 2022 07:41:00 %S A000086 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, %T A000086 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, %U A000086 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 %N A000086 Number of solutions to x^2 - x + 1 == 0 (mod n). %C A000086 Number of elliptic points of order 3 for Gamma_0(n). %C A000086 Equivalently, number of fixed points of Gamma_0(n) of type rho. %C A000086 Values are 0 or a power of 2. %C A000086 Shadow transform of central polygonal numbers A002061. - _Michel Marcus_, Jun 06 2013 %C A000086 Empirical: a(n) == A001615(n) (mod 3) for all natural numbers n. - _John M. Campbell_, Apr 01 2018 %C A000086 From _Jianing Song_, Jul 03 2018: (Start) %C A000086 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). %C A000086 Equivalently, number of solutions to x^2 + x + 1 == 0 (mod n). (End) %D A000086 Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101. %D A000086 Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (3). %H A000086 Christian G. Bower, Table of n, a(n) for n = 1..2000 %H A000086 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. %H A000086 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. %H A000086 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.] %H A000086 N. J. A. Sloane, Transforms. %F A000086 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 %F A000086 a(A226946(n)) = 0; a(A034017(n)) > 0. - _Reinhard Zumkeller_, Jun 23 2013 %F A000086 a(2*n) = a(3*n + 2) = a(9*n) = a(9*n + 6) = 0. - _Michael Somos_, Aug 14 2015 %F A000086 Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*sqrt(3)/(3*Pi) = 0.367552... (A165952). - _Amiram Eldar_, Oct 11 2022 %e A000086 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 + ... %p A000086 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 %t A000086 Array[ Function[ n, If[ EvenQ[ n ] || Mod[ n, 9 ]==0, 0, Count[ Array[ Mod[ #^2-#+1, n ]&, n, 0 ], 0 ] ] ], 84 ] %t A000086 a[ n_] := If[ n < 1, 0, Length[ Select[ (#^2 - # + 1)/n & /@ Range[n], IntegerQ]]]; (* _Michael Somos_, Aug 14 2015 *) %t A000086 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 *) %o A000086 (PARI) {a(n) = if( n<1, 0, sum( x=0, n-1, (x^2 - x + 1)%n==0))}; \\ Nov 15 2002 %o A000086 (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 %o A000086 (Haskell) %o A000086 a000086 n = if n `mod` 9 == 0 then 0 %o A000086 else product $ map ((* 2) . a079978 . (+ 2)) $ a027748_row $ a038502 n %o A000086 -- _Reinhard Zumkeller_, Jun 23 2013 %Y A000086 Cf. A000089, A000091, A001616, A014683. %Y A000086 Cf. A027748, A079978, A038502, A007949, A165952. %Y A000086 Cf. A341422 (without zeros). %K A000086 nonn,easy,nice,mult %O A000086 1,7 %A A000086 _N. J. A. Sloane_ # Content is available under The OEIS End-User License Agreement: http://oeis.org/LICENSE