A000086 -id:A000086 - cp's OEIS frontend

A341422 a(n) is the number of solutions of the congruence j^2 + j + 1 == 0 (mod k = A034017(n+1)), for j from {0, 1, 2, ..., k-1}, for n >= 1.

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

Offset: 1

Wolfdieter Lang, Apr 08 2021

This gives the row lengths of the irregular triangle A343232.
This sequence gives the number of representative parallel primitive forms (rpapfs) of the positive definite binary quadratic form F = x^2 + x*y + y^2 (with Discriminant Disc = -3) representing positive integers k. Only certain odd k, namely k = k(n) = A034017(n+1), for n >= 1, have proper solutions F = k.
Positive definite binary quadratic primitive forms F = [a, b, c], with a > 0 and gcd(a, b, c) = 1, with odd discriminants Disc = b^2 - 4*a*c = -D < 0, that is, D == 3 (mod 4), and representation of positive integers k have representative parallel primitive forms (rpapfs) Fpa(D,k;j) = [k, 2*j+1, (j^2 + j + (D+1)/4)/k].
Each rpapf produces a trivial proper solution to F = k, obtained from the trivial solution of Fpa(D,k;j) = k by (x, y) = (1,0), via equivalence transformations of determinant +1 achieved by applying the inverse of products of matrices R(t) = Mat([0,-1], [1t]]) for certain values t. The R(t) transformations are used to obtain from a primitive form F = [a, b, c] the equivalent so-called unique half-reduced (right) neighbor form F' = [c, -b + 2*c*t, a - b*t + c*t^2], with the choice t = ceiling((b/c - 1)/2). (c > 0 because a > 0 for positive definite forms with D > 0.)

Cf. A000086 (with zeros), A002061, A034017, A343232.

a(n) = |M(k(n))|, with the set M(k(n)) := {j from {0, 1, ..., k(n)-1} | j^2 + j + 1 == 0 (mod k(n))}, where j^2 + j + 1 = 2*T(j) + 1 = A002061(j+1) and k(n) = A034017(n+1), for n >= 1.

A353886 Nonnegative numbers k such that k^2 + k + 1 is squarefree.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 19, 20, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 34, 35, 36, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 69, 70, 71, 72

Offset: 1

Rémy Sigrist, May 09 2022

Dimitrov proved that this sequence is infinite.

The number of terms not exceeding X is Product_{p prime} (1 - A000086(p)/p^2) * X + O(X^(4/5+eps)) (Dimitrov, 2023). The coefficient of X, which is the asymptotic density of this sequence, equals Product_{primes p == 1 (mod 3)} (1 - 2/p^2) = 0.93484201367... . - Amiram Eldar, Dec 11 2023

			For k = 4, 4^2 + 4 + 1 = 21 = 3 * 7 is squarefree, so 4 belongs to this sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000
Stoyan Ivanov Dimitrov, Square-free values of n^2+n+1, Georgian Mathematical Journal, Vol. 30, No. 3 (2023), pp. 333-348; arXiv preprint, arXiv:2205.02488 [math.NT], 2022-2023.

Cf. A000086, A002061, A005117, A353887 (corresponding squarefree numbers).

Select[Range[0, 72], SquareFreeQ[#^2 + # + 1] &] (* Amiram Eldar, Dec 11 2023 *)

PARI
```
is(k) = issquarefree(k^2 + k + 1);
```

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

Original entry on oeis.org

49267, 74135, 94091, 96463, 102727, 107643, 118639, 138483, 145125, 181703, 182675, 208523, 221943, 237387, 240735, 245263, 255783, 267765, 269627, 272583, 277943, 280647, 283887, 286815, 309663, 313447, 322435, 326355, 336675, 347823, 352719

Offset: 1

Janos A. Csirik, Apr 21 2000

nonn

There are 4329 odd integers in the sequence less than 10^7. - Gheorghe Coserea, May 23 2016

J. A. Csirik, The genus of X_0(N) is not 150, preprint, 2000.

Gheorghe Coserea, Table of n, a(n) for n = 1..4329
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A054726, A054727, A054729.

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k, 1]+1)),
     h = prod(k=1, fsz, f[k, 1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
scan(n) = {
  my(inv = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n && inv[g+1] == -1, inv[g+1] = k));
  select(x->(x%2==1), apply(x->(x-1), Vec(select(x->x==-1, inv, 1))));
};
scan(400*1000)

More terms from Gheorghe Coserea, May 23 2016

Offset corrected by Gheorghe Coserea, May 23 2016

A087694 Number of solutions to x^2 + xy + y^2 == 0 (mod n).

Original entry on oeis.org

1, 1, 3, 4, 1, 3, 13, 4, 9, 1, 1, 12, 25, 13, 3, 16, 1, 9, 37, 4, 39, 1, 1, 12, 25, 25, 27, 52, 1, 3, 61, 16, 3, 1, 13, 36, 73, 37, 75, 4, 1, 39, 85, 4, 9, 1, 1, 48, 133, 25, 3, 100, 1, 27, 1, 52, 111, 1, 1, 12, 121, 61, 117, 64, 25, 3, 133, 4, 3, 13

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Sep 27 2003

Andrew Howroyd, Table of n, a(n) for n = 1..10000

Cf. A000086, A073010, A086724.

A087694 := proc(n) option remember; local pf,p,f,e ; if n = 1 then 1; else pf := ifactors(n)[2] ; if nops(pf) = 1 then f := op(1,pf) ; p := op(1,f) ; e := op(2,f) ; if p = 3 then n ; elif p mod 3 =1 then ((p-1)*e+p)*p^(e-1) ; else p^(2*floor(e/2)) ; end if; else mul(procname(op(1,p)^op(2,p)),p=pf) ; end if; end if; end proc:
seq(A087694(n),n=1..70) ; # R. J. Mathar, Jan 07 2011

a[n_] := If[n==1, 1, Product[{p, e} = pe; Which[p==3, 3^e, Mod[p, 3] == 2, (p^2)^Quotient[e, 2], True, ((p-1) e + p) p^(e-1)], {pe, FactorInteger[n] }]];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)

a(n)={my(f=factor(n)); prod(i=1, #f~, my(p=f[i,1], e=f[i,2]); if(p==3, 3^e, if(p%3==2, (p^2)^(e\2), ((p-1)*e+p)*p^(e-1))))} \\ Andrew Howroyd, Jul 09 2018

Multiplicative with a(3^e) = 3^e, a(p^e) = ((p-1)*e+p)*p^(e-1) if p mod 3 = 1, a(p^e) = p^(2*floor(e/2)) if p mod 3 = 2. - Vladeta Jovovic, Sep 27 2003

Sum_{k=1..n} a(k) ~ c * n^2 / 2, where c = A073010/A086724 = 0.77383581325017004332... . - Amiram Eldar, Nov 21 2023

A157227 Number of primitive inequivalent (up to Pi/3 rotation) non-hexagonal sublattices of hexagonal (triangular) lattice of index n.

Original entry on oeis.org

0, 1, 1, 2, 2, 4, 2, 4, 4, 6, 4, 8, 4, 8, 8, 8, 6, 12, 6, 12, 10, 12, 8, 16, 10, 14, 12, 16, 10, 24, 10, 16, 16, 18, 16, 24, 12, 20, 18, 24, 14, 32, 14, 24, 24, 24, 16, 32, 18, 30, 24, 28, 18, 36, 24, 32, 26, 30, 20, 48, 20, 32, 32, 32, 28, 48, 22, 36, 32, 48

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

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 4.]

Cf. A000086 (primitive hexagonal sublattices), A002324 (all hexagonal sublattices), A145394 (all sublattices), A001615, A304182.

a(n) = (A001615(n) - A000086(n))/3. - Andrey Zabolotskiy, May 09 2018

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

A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 0, 1, 2, 1, 2, 1, 3, 2, 2, 2, 5, 2, 4, 3, 5, 3, 4, 4, 6, 5, 6, 4, 10, 4, 6, 6, 8, 6, 10, 5, 9, 7, 8, 6, 14, 6, 10, 10, 11, 7, 12, 8, 14, 10, 12, 8, 17, 10, 12, 11, 14, 9, 20, 9, 15, 14, 14, 12, 22, 10, 16, 14, 22, 11, 20, 11, 18, 18, 18

Offset: 1

N. J. A. Sloane, Feb 25 2009

nonn

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 5.]

Cf. A003051 (all sublattices), A003050 (all primitive sublattices), A154272 (primitive sublattices fully inheriting the parent lattice symmetry, inlcuding the orientation of the mirrors), A000086 (primitive rotation-symmetric sublattices, counting mirror images as distinct), A060594 (primitive mirror-symmetric sublattices), A145377 (all sublattices inheriting the parent lattice symmetry), A304182.

a(n) = A003050(n) - (A000086(n)-A154272(n))/2 - A060594(n). - Andrey Zabolotskiy, Mar 19 2021

New name and a(1)=0 prepended by Andrey Zabolotskiy, May 09 2018

Terms a(31) and beyond from Andrey Zabolotskiy, Mar 19 2021

A224516 Number of solutions to x^4 - x == 0 (mod n).

Original entry on oeis.org

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

Offset: 1

Eric M. Schmidt, Apr 09 2013

			The solutions for n = 7 are 0, 1, 2, and 4.

Eric M. Schmidt, Table of n, a(n) for n = 1..10000

Cf. A034444, A087688, A060839, A000086.

f[3, e_] := If[e == 1, 2, 4]; f[p_, e_] := If[Mod[p, 3] == 2, 2, 4]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)

def A224516(n) :
    res = 1
    for p, m in factor(n) :
        if (p % 3 == 2) or (p == 3 and m == 1) : res *= 2
        else : res *= 4
    return res

Multiplicative with a(p^e) = 4 for p == 1 (mod 3); a(p^e) = 2 for p == 2 (mod 3); a(3^1) = 2; a(3^e) = 4 for e > 1.

A273445 a(n) is the number of solutions of the equation n = A001617(k).

Original entry on oeis.org

15, 12, 8, 11, 7, 14, 4, 13, 7, 12, 4, 15, 4, 9, 6, 10, 5, 16, 2, 20, 3, 14, 7, 11, 2, 13, 5, 11, 3, 14, 3, 9, 6, 13, 3, 17, 3, 14, 4, 10, 4, 20, 3, 15, 3, 12, 1, 15, 2, 20, 4, 11, 3, 13, 3, 16, 3, 12, 3, 15, 3, 12, 5, 9, 4, 15, 2, 14, 5, 17, 3, 13

Offset: 0

Gheorghe Coserea, May 22 2016

nonn

The zeros of the sequence are given by A054729. The first five zeros of the sequence have indexes 150, 180, 210, 286, 304.

			For n = 0 the a(0) = 15 solutions are:
1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401).
For n = 1 the a(1) = 12 solutions are:
11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403).
For n = 2 the a(2) = 8 solutions are:
22, 23, 26, 28, 29, 31, 37, 50 (A091404).

Gheorghe Coserea, Table of n, a(n) for n = 0..100001
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054729, A091401, A091403, A091404.

(* b = A001617 *) nmax = 71;
b[n_] := b[n] = If[n < 1, 0, 1 + Sum[ MoebiusMu[ d]^2 n/d / 12 - EulerPhi[ GCD[ d, n/d]] / 2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 -Count[(#^2 + 1)/n& /@ Range[n], ?IntegerQ]/4];
Clear[f];
f[m_] := f[m] = Module[{}, A001617 = Array[b, m]; a[n_] := Count[A001617, n]; Table[a[n], {n, 0, nmax}]];
f[m = nmax]; f[m = m + nmax];
While[Print["m = ", m]; f[m] != f[m - nmax], m = m + nmax];
A273445 = f[m] (* Jean-François Alcover, Dec 16 2018, using Michael Somos' code for A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k,1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k,1]+1)),
     h = prod(k=1, fsz, f[k,1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k,1]^(f[k,2]\2) + f[k,1]^((f[k,2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
seq(n) = {
  my(a = vector(n+1,g,0), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k);
       if (g <= n, a[g+1]++));
  return(a);
};
seq(72)

a(n) = card {k, n = A001617(k)}.

A087781 Number of non-congruent solutions to x^2 - x - 1 == 0 mod n.

Original entry on oeis.org

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

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003

Sequence A089270 gives the positions of the nonzero terms. The term a(n) gives the number of primitive solutions (x,y) of the equation x^2 + xy - y^2 = n.

T. D. Noe, Table of n, a(n) for n=1..10000

Cf. A000086, A034444.

More terms from David Wasserman, Jun 17 2005

A273510 a(n) is the largest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

Original entry on oeis.org

25, 49, 50, 64, 81, 75, 121, 100, 169, 128, 127, 147, 157, 163, 181, 193, 199, 289, 229, 243, 239, 257, 361, 283, 293, 313, 343, 337, 349, 353, 373, 379, 397, 409, 421, 529, 439, 457, 463, 467, 487, 499, 509, 523, 541, 547, 557, 577, 625, 601, 613, 619, 631, 643, 661, 673, 677, 691, 841, 667, 733

Offset: 0

Gheorghe Coserea, May 23 2016

sign

a(10^7) = 120000007 is the largest value in the first 1+10^7 terms of the sequence.

The exception occurs first at a(150) = -1. - Georg Fischer, Feb 15 2019

			For n = 0 we have 0 = A001617(k) when k is 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 13, 16, 18, 25 (A091401); the largest of this values is 25 therefore a(0) = 25.
For n = 1 we have 1 = A001617(k) when k is 11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49 (A091403); the largest of this values is 49 therefore a(1) = 49.
For n = 2 we have 2 = A001617(k) when k is 22, 23, 26, 28, 29, 31, 37, 50 (A091404); the largest of this values is 50 therefore a(2) = 50.
For n = 150 (= A054729(1)) we have 150 <> A001617(k) for all k therefore a(150) = -1.

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
J. A. Csirik, M. Zieve, and J. Wetherell, On the genera of X0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054728, A054729, A091401, A091403, A091404, A273445.

a1617[n_] := If[n < 1, 0, 1 + Sum[MoebiusMu[d]^2 n/d/12 - EulerPhi[GCD[d, n/d]]/2, {d, Divisors[n]}] - Count[(#^2 - # + 1)/n& /@ Range[n], ?IntegerQ]/3 - Count[(#^2 + 1)/n& /@ Range[n], ?IntegerQ]/4];
seq[n_] := Module[{a, bnd}, a = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n] ] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n, a[[g+1]] = k]]; a];
seq[60] (* Jean-François Alcover, Nov 20 2018, after Gheorghe Coserea and Michael Somos in A001617 *)

A000089(n) = {
  if (n%4 == 0 || n%4 == 3, return(0));
  if (n%2 == 0, n \= 2);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 4 == 3, 0, 2));
};
A000086(n) = {
  if (n%9 == 0 || n%3 == 2, return(0));
  if (n%3 == 0, n \= 3);
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, if (f[k, 1] % 3 == 2, 0, 2));
};
A001615(n) = {
  my(f = factor(n), fsz = matsize(f)[1],
     g = prod(k=1, fsz, (f[k, 1]+1)),
     h = prod(k=1, fsz, f[k, 1]));
  return((n*g)\h);
};
A001616(n) = {
  my(f = factor(n), fsz = matsize(f)[1]);
  prod(k = 1, fsz, f[k, 1]^(f[k, 2]\2) + f[k, 1]^((f[k, 2]-1)\2));
};
A001617(n) = 1 + A001615(n)/12 - A000089(n)/4 - A000086(n)/3 - A001616(n)/2;
seq(n) = {
  my(a = vector(n+1, g, -1), bnd = 12*n + 18*sqrtint(n) + 100, g);
  for (k = 1, bnd, g = A001617(k); if (g <= n, a[g+1] = k));
  return(a);
};
seq(60)

Let S(n) = {k, n = A001617(k)}; if the level set S(n) is not empty then a(n) = max S(n) and A054728(n) = min S(n) and A273445(n) = card S(n), otherwise a(n) = A054728(n) = -1 and A273445(n) = 0.

cp's OEIS Frontend

A341422 a(n) is the number of solutions of the congruence j^2 + j + 1 == 0 (mod k = A034017(n+1)), for j from {0, 1, 2, ..., k-1}, for n >= 1.

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A353886 Nonnegative numbers k such that k^2 + k + 1 is squarefree.

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PARI

A054730 Odd n such that genus of modular curve X_0(N) is never equal to n.

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A087694 Number of solutions to x^2 + xy + y^2 == 0 (mod n).

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Maple

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PARI

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A157227 Number of primitive inequivalent (up to Pi/3 rotation) non-hexagonal sublattices of hexagonal (triangular) lattice of index n.

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A157235 Number of primitive inequivalent oblique sublattices of hexagonal (triangular) lattice of index n (equivalence and symmetry of sublattices are determined using only parent lattice symmetries).

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A224516 Number of solutions to x^4 - x == 0 (mod n).

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Mathematica

Sage

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A273445 a(n) is the number of solutions of the equation n = A001617(k).

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