A060594 -id:A060594 - cp's OEIS frontend

A319101 Number of solutions to x^7 == 1 (mod n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 7

Offset: 1

Jianing Song, Sep 10 2018

All terms are powers of 7. Those n such that a(n) > 1 are in A066502.

			Solutions to x^7 == 1 (mod 29): x == 1, 7, 16, 20, 23, 24, 25 (mod 29).
Solutions to x^7 == 1 (mod 43): x == 1, 4, 11, 16, 21, 35, 41 (mod 43).
Solutions to x^7 == 1 (mod 49): x == 1, 8, 15, 22, 29, 36, 43 (mod 49) (x == 1 (mod 7)).

Jianing Song, Table of n, a(n) for n = 1..10000

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), this sequence (k=7), A247257 (k=8).
Cf. A066502, A140444, A293484, A000010.
Mobius transform gives A307382.

f[p_, e_] := If[Mod[p, 7] == 1, 7, 1]; f[7, 1] = 1; f[7, e_] := 7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 10 2023 *)

a(n)=my(Z=znstar(n)[2]); prod(i=1, #Z, gcd(7, Z[i]))

Multiplicative with a(7) = 1, a(7^e) = 7 if e >= 2; for other primes p, a(p^e) = 7 if p == 1 (mod 7), a(p^e) = 1 otherwise.
If the multiplicative group of integers modulo n is isomorphic to C_{k_1} x C_{k_2} x ... x C_{k_m}, where k_i divides k_j for i < j; then a(n) = Product_{i=1..m} gcd(7, k_i).
a(n) = A000010(n)/A293484(n). - Jianing Song, Nov 10 2019

A114643 Number of real primitive Dirichlet characters modulo n.

Original entry on oeis.org

1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 1, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 0, 1, 2, 0, 0, 0, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 2, 1, 0, 1, 1, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 0, 0, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 1, 1, 0, 1, 2, 1, 0, 1, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 2, 1

Offset: 1

Steven Finch, Feb 16 2006

a(n) = 1 if either n or -n is a fundamental discriminant (not both); a(n) = 2 if n and -n are fundamental discriminants; a(n) = 0 otherwise. Also, Sum_{k=1..n} a(k) is asymptotic to (6/Pi^2)*n.
From Jianing Song, Feb 27 2019: (Start)
If n is an odd squarefree number, then a(n) = 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(n,k)} = {Jacobi(k,n)} if n == 1 (mod 4) and {Kronecker(-n,k)} = {Jacobi(k,n)} if n == 3 (mod 4).
If n = 4*m, m is an odd squarefree number, then a(n) is also 1, where the unique real primitive Dirichlet character modulo n is {Kronecker(-n,k)} if m == 1 (mod 4) and {Kronecker(n,k)} if m == 3 (mod 4).
If n is 8 times an odd squarefree number, then a(n) = 2, where the two real primitive Dirichlet characters modulo n are {Kronecker(n,k)} and {Kronecker(-n,k)}.
a(n) = 0 if n == 2 (mod 4), n is divisible by 16 or the square of an odd prime. (End)
Mobius transform of A060594. - Jianing Song, Mar 02 2019

			From _Jianing Song_, Feb 27 2019: (Start)
For n = 5, the only real primitive Dirichlet characters modulo n is {Kronecker(5,k)} = [0, 1, -1, -1, 1] = A080891, so a(5) = 1.
For n = 8, the real primitive Dirichlet characters modulo n are {Kronecker(8,k)} = [0, 1, 0, -1, 0, -1, 0, 1] = A091337 and [0, 1, 0, 1, 0, -1, 0, -1] = A188510, so a(8) = 2.
For n = 20, the only real primitive Dirichlet characters modulo n is {Kronecker(-20,k)} = [0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1] = A289741, so a(20) = 1. (End)

W. Ellison and F. Ellison, Prime Numbers, Wiley, 1985, pp. 224-226.

Robert Israel, Table of n, a(n) for n = 1..10000
Steven R. Finch, Cubic and quartic characters [Broken link]
Steven R. Finch, Cubic and quartic characters.
Vaclav Kotesovec, Graph - the asymptotic ratio
Eric Weisstein's World of Mathematics, Dirichlet L-Series.
I. J. Zucker and M. M. Robertson, Some properties of Dirichlet L-series, J. Phys. A 9 (1976) 1207-1214.

Cf. A003657, A003658.
Cf. A160498 (number of cubic primitive Dirichlet characters modulo n), A160499 (number of quartic primitive Dirichlet characters modulo n).
Cf. A060594 (number of solutions to x^2 == 1 (mod n)).

A114643 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            if r =  1 then
                a := 0 ;
            elif r =  2 then
                ;
            elif r =  3 then
                a := a*2 ;
            elif r >=  4 then
                a := 0 ;
            end if;
        else
            if r =1 then
                ;
            else
                a := 0 ;
            end if;
        end if;
    end do:
    a ;
end proc:
seq(A114643(n),n=1..40) ; # R. J. Mathar, Mar 02 2015
# Alternative:
f:= proc(n) local r,v,F;
  v:= padic:-ordp(n,2);
  if v = 1 or v >= 4 then return 0
  elif v = 3 then r:= 2
  else r:= 1
  fi;
  if numtheory:-issqrfree(n/2^v) then r else 0 fi
end proc:
map(f, [$1..100]); # Robert Israel, Oct 08 2017

a[n_] := Sum[ MoebiusMu[n/d] * Sum[ If[ Mod[i^2 - 1, d] == 0, 1, 0], {i, 2, d}], {d, Divisors[n]}]; a[1] = 1; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Jun 20 2013, after Steven Finch *)
f[2, e_] := Which[e == 1, 0, e == 2, 1, e == 3, 2, e >= 4, 0]; f[p_, e_] := If[e == 1, 1, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 16 2020 *)

a(n)=sum(d=1, n, if(n%d==0, moebius(n/d)*sum(i=1, d, if((i^2-1)%d, 0, 1)), 0)) \\ Steven Finch, Jun 09 2009

This sequence is multiplicative with a(2) = 0, a(4) = 1, a(8) = 2, a(2^r) = 0 for r > 3, a(p) = 1 for prime p > 2 and a(p^r) = 0 for r > 1. - Steven Finch, Mar 08 2006 (With correction by Jianing Song, Jun 28 2018)

Dirichlet g.f.: zeta(s)*(1 + 2^(-2s) + 2^(1-3s))/(zeta(2s)*(1 + 2^(-s))). - R. J. Mathar, Jul 03 2011

A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

Original entry on oeis.org

1, 2, 3, 5, 4, 7, 5, 10, 8, 10, 7, 17, 8, 13, 14, 19, 10, 21, 11, 24, 18, 19, 13, 35, 17, 22, 22, 31, 16, 38, 17, 36, 26, 28, 26, 50, 20, 31, 30, 50, 22, 50, 23, 45, 42, 37, 25, 69, 30, 48, 38, 52, 28, 62, 38, 65, 42, 46, 31, 90, 32, 49, 55, 69, 44, 74, 35, 66, 50, 74

Offset: 1

N. J. A. Sloane, Feb 23 2009

nonn

The centered rectangular lattice has symmetry group c2mm, or cmm. For other 2D Patterson groups, the analogous sequences are A000203 (p2), A069734 (p2mm), A145392 (p4), A145393 (p4mm), A145394 (p6), A003051 (p6mm). - Andrey Zabolotskiy, Mar 12 2018

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] [see Table 8].
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 2.]
Index entries for sequences related to sublattices

Cf. A000203, A069734, A145390, A145392, A145393, A145394, A003051, A060594, A157223, A004525.

a060594[n_] := Switch[Mod[n, 8], 2|6, 2^(PrimeNu[n] - 1), 1|3|4|5|7, 2^PrimeNu[n], 0, 2^(PrimeNu[n] + 1)];
a145390[n_] := Sum[If[IntegerQ[Sqrt[d]], a060594[n/d], 0], {d, Divisors[n]} ];
a[n_] := (DivisorSigma[1, n] + a145390[n])/2;
Array[a, 100] (* Jean-François Alcover, Aug 31 2018 *)

a(n) = (A000203(n) + A145390(n))/2. [Rutherford] - N. J. A. Sloane, Mar 13 2009
a(n) = Sum_{ m: m^2|n } A060594(n/m^2) + A157223(n/m^2) = A145390(n) + Sum_{ m: m^2|n } A157223(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = Sum_{ d|n } A004525(d+1). - Andrey Zabolotskiy, Aug 29 2019

New name from Andrey Zabolotskiy, Mar 12 2018

New name from Andrey Zabolotskiy, Jan 19 2022

A247257 The number of octic characters modulo n.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 2, 4, 2, 4, 2, 4, 4, 2, 8, 8, 8, 2, 2, 8, 4, 2, 2, 8, 4, 4, 2, 4, 4, 8, 2, 16, 4, 8, 8, 4, 4, 2, 8, 16, 8, 4, 2, 4, 8, 2, 2, 16, 2, 4, 16, 8, 4, 2, 8, 8, 4, 4, 2, 16, 4, 2, 4, 16, 16, 4, 2, 16, 4, 8, 2, 8, 8, 4, 8, 4, 4, 8, 2, 32

Offset: 1

R. J. Mathar, Mar 02 2015

Number of solutions to x^8 == 1 (mod n). - Jianing Song, Nov 10 2019

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Steven Finch, Quartic and octic characters modulo n, arXiv:0907.4894 [math.NT], 2009.

Number of solutions to x^k == 1 (mod n): A060594 (k=2), A060839 (k=3), A073103 (k=4), A319099 (k=5), A319100 (k=6), A319101 (k=7), this sequence (k=8).

A247257 := proc(n)
    local a,pf,p,r;
    a := 1 ;
    for pf in ifactors(n)[2] do
        p := op(1,pf);
        r := op(2,pf);
        if p = 2 then
            if r >= 5 then
                a := a*16 ;
            else
                a := a*op(r,[1,2,4,8]) ;
            end if;
        elif modp(p,4) = 3 then
            a := a*2;
        elif modp(p,8) = 5 then
            a := a*4;
        elif modp(p,8) = 1 then
            a := a*8;
        else
            error
        end if;
    end do:
    a ;
end proc:

g[p_, e_] := Which[p==2, 2^Min[e-1, 4], Mod[p, 4]==3, 2, Mod[p, 8]==5, 4, True, 8];
a[1] = 1; a[n_] := Times @@ g @@@ FactorInteger[n];
Array[a, 80] (* Jean-François Alcover, Nov 26 2017, after Charles R Greathouse IV *)

g(p,e)=if(p==2, 2^min(e-1,4), if(p%4==3, 2, if(p%8==5, 4, 8)))
a(n)=my(f=factor(n)); prod(i=1,#f~, g(f[i,1],f[i,2])) \\ Charles R Greathouse IV, Mar 02 2015

Multiplicative with a(p^e) = p^min(e-1, 4) if p = 2, gcd(8, p-1) if p > 2. - Jianing Song, Nov 10 2019

A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 2, 2, 1, 1, 2, 4, 1, 2, 1, 2, 2, 1, 1, 4, 1, 1, 2, 2, 1, 1, 2, 4, 2, 1, 1, 4, 1, 1, 2, 2, 2, 2, 1, 2, 2, 2, 1, 4, 1, 1, 2, 2, 2, 2, 1, 4, 1, 1, 1, 4, 2, 1, 2

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

After a(2), this matches A034380 except for n = 63, 65, 80, 85, ... - R. J. Mathar, Feb 27 2009 [Updated by Andrey Zabolotskiy, May 09 2018]

Andrey Zabolotskiy, Table of n, a(n) for n = 1..5000
J. S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Act. Cryst. A65 (2009) 156-163, Table 5 symmetry *mm2.

Cf. A145393 (all sublattices of the square lattice), A019590, A157228, A157226, A157231, A304182, A060594, A046072, A033948, A272592.

a[n_] := If[n <= 2, 0, Sum[Boole[Mod[k^2, n] == 1], {k, 1, n}]/2];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Apr 12 2023 *)

From Andrey Zabolotskiy, Sep 30 2018: (Start)
a(n) = (A060594(n) - A019590(n))/2.
a(n) = 2^(A046072(n)-1) for n>2. Thus a(n) = 1 if n>2 is in A033948, a(n) = 2 if n is in A272592, etc. (End)

New name and more terms from Andrey Zabolotskiy, May 09 2018

A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ.

Original entry on oeis.org

1, 6, 48, 384, 240, 288, 672, 6144, 1296, 1440, 2640, 18432, 4368, 4032, 11520, 49152, 9792, 7776, 13680, 92160, 32256, 15840, 24288, 294912, 30000, 26208, 34992, 258048, 48720, 69120, 59520, 393216, 126720, 58752, 161280, 497664, 101232, 82080, 209664, 1474560

Offset: 1

Charles Repizo, Nov 03 2015

Number of matrices M = [a,b,c; d,e,f; g,h,i] with 0 <= a, b, c, d, e, f, g, h, i < n such that M*transpose(M) == [1,0,0; 0,1,0; 0,0,1] (mod n).

For n > 1, a(n) is divisible by 6*A060594(n)^3. - Robert Israel, Dec 16 2015

Robert Israel, Table of n, a(n) for n = 1..124

Cf. A060594, A087784, A208895.

Enter R := IntegerRing(n);
korthmat := function(R,n,k);
O := [];
M := MatrixAlgebra(R,n);
for x in M do
if x*Transpose(x) eq k*M!1 and Transpose(x)*x eq k*M!1 then
O := Append(O,x);
end if;
end for;
return O;
end function;
# korthmat(R,3,1);

F:= proc(n) local R,V,nR,S,nS,Rp,nRp,i,j,a,b,c,t,r,r1,count;
      R:= select(t -> t[1]^2 + t[2]^2 + t[3]^2 mod n = 1, [seq(seq(seq([a,b,c],a=0..n-1),b=0..n-1),c=0..n-1)]);
      nR:= nops(R);
      S:= select(t -> t^2 mod n = 1, {$2..n-1});
      nS:= nops(S);
      for r in R do if not assigned(V[r]) then
         for c in S do V[c*r mod n] := 0 od
      fi od;
      R:= select(r -> not assigned(V[r]), R);
      nR:= nops(R);
      count:= 0;
      for i from 1 to nR do
        r:= R[i];
        Rp:= select(j -> R[j][1]*r[1] + R[j][2]*r[2] + R[j][3]*r[3] mod n = 0, [$i+1..nR]);
        nRp:= nops(Rp);
        for j from 1 to nRp do
            r1:= R[Rp[j]];
            count:= count + 6*(1+nS)^3*nops(select(k -> R[Rp[k]][1]*r1[1] + R[Rp[k]][2]*r1[2]+R[Rp[k]][3]*r1[3] mod n = 0, [$j+1..nRp]));
        od
      od;
      count;
end proc:
F(1):= 1:
seq(F(n), n=1..40); # Robert Israel, Dec 16 2015

my(t=Mod(matid(3), n)); sum(a=1, n, sum(b=1, n, sum(c=1, n, sum(d=1, n, sum(e=1, n, sum(f=1, n, sum(g=1, n, sum(h=1, n, sum(i=1, n, my(M=[a, b, c; d, e, f; g, h, i]); M*M~==t))))))))) \\ Charles R Greathouse IV, Nov 10 2015

For p an odd prime, a(p) = 2*p*(p^2-1). - Tom Edgar, Nov 04 2015
From Robert Israel, Dec 16 2015: (Start)
Conjectures:
  a(2^k) = 12*8^k for k >= 3.
  For odd primes p, a(p^k) = a(p)*p^(3k-3) for k>=1. (End)

a(11)-a(31) from Tom Edgar, Nov 05 2015

a(31) corrected by Robert Israel, Dec 15 2015

A141453 Primes p such that either p = 2^k + 1 or p = 2^k - 1, k>=0.

Original entry on oeis.org

2, 3, 5, 7, 17, 31, 127, 257, 8191, 65537, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Leroy Quet, Aug 07 2008

nonn

Sequence consists of 2 and the union of the Mersenne primes (A000668) and the Fermat primes (A019434).
a(18) has 157 digits and is too large to include. - Ray Chandler, Jun 22 2009
From Wolfdieter Lang, Mar 28 2012: (Start)
The sequence of exponents k of 2 for these Mersenne or Fermat primes is k=[0, 1, 2, 3, 4, 5, 7, 8, 13, 16, 17, 19, 31, 61, 89, 107, 127,...], n>=1, if 3 is considered as Fermat prime.
  The second exponent is 2 if 3 is considered as Mersenne prime. The next exponent k(18)=521 for the Mersenne prime a(18).
r(a(n)) := sqrt(a(n)*2^(k(n)+2) + 1) is a solution of the congruence x^2 == 1 (mod a(n)*2^(k(n)+1)), n>=1. The sequence k is given in the preceding comment. If n=2 then, besides r(3) = 5, also sqrt(3*2^(2+2) + 1) = 7 is an incongruent solution mod 12 if the exponent 2 is chosen. For n=2 there are altogether four incongruent solutions of x^2 == 1 (mod 12), namely 1, 12-1 = 11, r(3)=5 and 12-5 = 7. If n=1 there are altogether two incongruent solutions, namely 1 and r(2) = 3 = 4-1 (a trivial solution == -1 (mod 4)). For n>=3 there are eight incongruent solutions, and besides the trivial (positive) ones, 1 and a(n)*2^(k(n)+1) - 1, one has a nontrivial pair r(a(n)) and a(n)*2^(k(n)+1) - r(a(n)). For the number of incongruent solutions of the congruence x^2 == 1 (mod n) see A060594. For the r(a(n)) values see A210844.
r(a(n)), n>=1, also solves the congruence x^2 == 1 (Modd a(n)*2^(k(n)+1)), because floor((r(a(n))^2)/(a(n)*2^(k(n)+1)) is even. For Modd n (not to be confused with mod n) see a comment on A203571.
(End)

			From _Wolfdieter Lang_, Mar 28 2012: (Start)
Solutions to the congruence x^2 == 1 (mod a(n)*2^(k(n)+1):
n=3: r(5) = sqrt(5*2^(2+2) + 1) = 9. 9^2 = 81 == 1 (mod 5*8).
  The companion solution is 40-9 = 31. Because floor(81/40)=2 is even, 81 == 1 (Modd 40) also.
n=4: r(7) =  sqrt(7*2^(3+2) + 1) = 15. 15^2 = 225 == 1 (mod 7*16). The companion solution is 112-15 = 97. Because floor(225/112)=2 is even, 225 == 1 (Modd 112) also.
n=7: r(127) = sqrt(127*2^(7+2) + 1) = 255. 255^2 == 1 (mod 127*2^8). The companion solution is 32512-255 = 32257.
(End)

Michael De Vlieger, Table of n, a(n) for n = 1..22

Cf. A000668, A019434.

Select[Prime[Range[30000]], Length[FactorInteger[#-1]]==1 || Length[FactorInteger[#+1]]==1&] (* Vladimir Joseph Stephan Orlovsky, Feb 03 2012 *)
Select[Union[Join @@ Array[2^# + {-1, +1} &, 140, 0]], PrimeQ] (* Michael De Vlieger, Oct 23 2017 *)

More terms from R. J. Mathar, Jan 23 2009

a(17) from Ray Chandler, Jun 22 2009

A145390 Number of sublattices of index n of a centered rectangular lattice fixed by a reflection.

Original entry on oeis.org

1, 1, 2, 3, 2, 2, 2, 5, 3, 2, 2, 6, 2, 2, 4, 7, 2, 3, 2, 6, 4, 2, 2, 10, 3, 2, 4, 6, 2, 4, 2, 9, 4, 2, 4, 9, 2, 2, 4, 10, 2, 4, 2, 6, 6, 2, 2, 14, 3, 3, 4, 6, 2, 4, 4, 10, 4, 2, 2, 12, 2, 2, 6, 11, 4, 4, 2, 6, 4, 4, 2, 15, 2, 2, 6, 6, 4, 4, 2, 14, 5, 2, 2, 12, 4, 2, 4, 10, 2, 6, 4, 6, 4, 2, 4, 18, 2, 3, 6, 9, 2

Offset: 1

N. J. A. Sloane, Feb 23 2009, Mar 13 2009

a(n) is the Dirichlet convolution of A000012 and A098178. - Domenico (domenicoo(AT)gmail.com), Oct 21 2009

Andrey Zabolotskiy, Table of n, a(n) for n = 1..10000
Amihay Hanany, Domenico Orlando, and Susanne Reffert, Sublattice counting and orbifolds, High Energ. Phys., 2010 (2010), 51, arXiv.org:1002.2981 [hep-th] (see Table 3).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009

Cf. A098178, A060594 (primitive sublattices only), A145391.

nmax := 100 :
L := [1,-1,0,2,seq(0,i=1..nmax)] :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

m = 101; Drop[ CoefficientList[ Series[ Sum[(1 + Cos[n*Pi/2])*x^n/(1 - x^n), {n, 1, m}], {x, 0, m}], x], 1] (* Jean-François Alcover, Sep 20 2011, after formula *)
f[p_, e_] := e+1; f[2, e_] := 2*e-1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 27 2023 *)

t1=direuler(p=2,200,1/(1-X)^2)
t2=direuler(p=2,2,1-X+2*X^2,200)
t3=dirmul(t1,t2)

Dirichlet g.f.: (1-2^(-s) + 2*4^(-s))*zeta^2(s).
G.f.: Sum_n (1 + cos(n*Pi/2)) x^n / (1 - x^n). - Domenico (domenicoo(AT)gmail.com), Oct 21 2009
If 4|n then a(n) = d(n) - d(n/2) + 2*d(n/4); else if 2|n then a(n) = d(n) - d(n/2); else a(n) = d(n); where d(n) is the number of divisors of n. [Rutherford] - Andrey Zabolotskiy, Mar 10 2018
a(n) = Sum_{ m: m^2|n } A060594(n/m^2). - Andrey Zabolotskiy, May 07 2018
Sum_{k=1..n} a(k) ~ n*(log(n) - 1 + 2*gamma - log(2)/2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 02 2019
Multiplicative with a(2^e) = 2*e-1 and a(p^e) = e+1 for an odd prime p. - Amiram Eldar, Aug 27 2023

New name from Andrey Zabolotskiy, Mar 10 2018

A180783 Number of distinct solutions of Sum_{i=1..1} (x(2i-1)*x(2i)) == 1 (mod n), with x() in {1,2,...,n-1}.

Original entry on oeis.org

0, 1, 2, 2, 3, 2, 4, 4, 4, 3, 6, 4, 7, 4, 6, 6, 9, 4, 10, 6, 8, 6, 12, 8, 11, 7, 10, 8, 15, 6, 16, 10, 12, 9, 14, 8, 19, 10, 14, 12, 21, 8, 22, 12, 14, 12, 24, 12, 22, 11, 18, 14, 27, 10, 22, 16, 20, 15, 30, 12, 31, 16, 20, 18, 26, 12, 34, 18, 24, 14, 36, 16, 37, 19, 22, 20, 32, 14, 40, 20, 28

Offset: 1

R. H. Hardin, formula from Max Alekseyev in the Sequence Fans Mailing List, Sep 20 2010

nonn

Except for the first term, this appears to be the number of pairs of integers i,j with 1 <= i <= n, 1 <= j <= i, such that i+j == i*j (mod n), for n=1,2,3,...  - John W. Layman,  Oct 19 2011
Layman's observation holds since i+j == i*j (mod n) is equivalent to (i-1)*(j-1) == 1 (mod n). - Max Alekseyev, Oct 22 2011
For i > 1, equal to the number of elements x relatively prime to n such that x mod n >= x^(-1) mod n. - Jeffrey Shallit, Jun 14 2018
Differs from A007897 for n = 1, 35, 45 etc. - Georg Fischer, Sep 20 2020

			Solutions for product of a single 1..10 pair = 1 (mod 11) are (1*1) (2*6) (3*4) (5*9) (7*8) (10*10).

R. H. Hardin, Table of n, a(n) for n = 1..10000

Column 1 of A180793.

Cf. A000010, A007897, A060594.

a(n) = (A000010(n) + A060594(n)) / 2.

A227091 Number of solutions to x^2 == 1 (mod n) in Z[i]/nZ[i].

Original entry on oeis.org

1, 2, 2, 4, 4, 4, 2, 8, 2, 8, 2, 8, 4, 4, 8, 8, 4, 4, 2, 16, 4, 4, 2, 16, 4, 8, 2, 8, 4, 16, 2, 8, 4, 8, 8, 8, 4, 4, 8, 32, 4, 8, 2, 8, 8, 4, 2, 16, 2, 8, 8, 16, 4, 4, 8, 16, 4, 8, 2, 32, 4, 4, 4, 8, 16, 8, 2, 16, 4, 16, 2, 16, 4, 8, 8, 8, 4, 16, 2, 32, 2, 8

Offset: 1

José María Grau Ribas, Jun 30 2013

Number of non-congruent solutions of x^2 + y^2 -1 == 2xy == 0 (mod n).
This sequence combines A329586 (number of representative solutions of a^2 - (b^2 + 1) == 0 (mod m) and 2*a*b == 0 (mod m) with a*b = 0), and those from A329589 (number of representative solutions of these two congruences but with a*b not 0). - Wolfdieter Lang, Dec 14 2019
In A226746 the positive n numbers with more than two representative solutions of the congruence z^2 = +1 (mod n) are given. This is therefore a proper subsequence of the present one. - Wolfdieter Lang, Dec 14 2019

			a(4) = 4 because in Z[i]/4Z[i] the equation x^2==1 (mod 4) has 4 solutions: 1, 1+2i, 3 and 3+2i.

Eric M. Schmidt, Table of n, a(n) for n = 1..1000

Cf. A060594, A226746, A329586, A329589.

a:= n-> mul(`if`(i[1]=2, 2^min(i[2], 3), `if`(
    irem(i[1], 4)=1, 4, 2)), i=ifactors(n)[2]):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 07 2020

h[n_] := Flatten[Table[a + b I, {a, 0, n - 1}, {b, 0, n - 1}]]; a[1] = 1; a[n_] := Length@Select[h[n], Mod[#^2, n] == 1 &]; Table[a[n], {n, 2, 44}]
f[2, e_] := 2^Min[e, 3]; f[p_, e_] := If[Mod[p, 4] == 1, 4, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

a(n)=my(o=valuation(n,2),f=factor(n>>o)[,1]); prod(i=1,#f, if(f[i]%4==1, 4, 2))<Charles R Greathouse IV, Dec 13 2013

def A227091(n) : return prod([4,2^min(m,3),2][p%4-1] for (p,m) in factor(n)) # Eric M. Schmidt, Jul 09 2013

Multiplicative with a(2^e) = 2^min(e, 3); a(p^e) = 4 for p == 1 (mod 4); a(p^e) = 2 for p == 3 (mod 4). - Eric M. Schmidt, Jul 09 2013

cp's OEIS Frontend

A319101 Number of solutions to x^7 == 1 (mod n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A114643 Number of real primitive Dirichlet characters modulo n.

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A145391 Number of inequivalent sublattices of index n in centered rectangular lattice.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A247257 The number of octic characters modulo n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A157230 Number of primitive inequivalent sublattices of square lattice having mirrors parallel to the diagonals of the unit cell of the parent lattice of index n.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

Formula

Extensions

A264083 Number of orthogonal 3 X 3 matrices over the ring Z/nZ.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Magma

Maple

PARI

Formula

Extensions

A141453 Primes p such that either p = 2^k + 1 or p = 2^k - 1, k>=0.

Views

Author

Keywords

Comments

Examples