A000089 -id:A000089 - cp's OEIS frontend

A091403 Numbers n such that genus of group Gamma_0(n) is 1.

Original entry on oeis.org

11, 14, 15, 17, 19, 20, 21, 24, 27, 32, 36, 49

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

Available conductors for modular elliptic curves genus 1. [From Artur Jasinski, Jun 24 2010]

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091404.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a86[n_] := a86[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, a86[p^e]], {pe, FactorInteger[n]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[ d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 1] // Flatten (* Jean-François Alcover, Oct 18 2018 *)

Numbers n such that A001617(n) = 1.

A054728 a(n) is the smallest 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

1, 11, 22, 30, 38, 42, 58, 60, 74, 66, 86, 78, 106, 105, 118, 102, 134, 114, 223, 132, 166, 138, 188, 156, 202, 168, 214, 174, 236, 186, 359, 204, 262, 230, 278, 222, 298, 240, 314, 246, 326, 210, 346, 270, 358, 282, 557, 306, 394, 312, 412, 318

Offset: 0

Janos A. Csirik, Apr 21 2000

sign

a(150) = -1, a(n) > 0 for 0<=n<=149.

a(9999988) = 119999861 is the largest value in the first 1+10^7 terms of the sequence. - Gheorghe Coserea, May 24 2016

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

Gheorghe Coserea, Table of n, a(n) for n = 0..200010
János A. Csirik, Joseph L. Wetherell, Michael E. Zieve, On the genera of X_0(N), arXiv:math/0006096 [math.NT], 2000.

Cf. A001617, A054727, A054729.

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[{inv, bnd}, inv = Table[-1, {n+1}]; bnd = 12n + 18 Floor[Sqrt[n]] + 100; For[k = 1, k <= bnd, k++, g = a1617[k]; If[g <= n && inv[[g+1]] == -1, inv[[g+1]] = k]]; inv];
seq[51] (* 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(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));
  return(inv);
};
seq(51)  \\ Gheorghe Coserea, May 21 2016

A001617(a(A001617(n))) = A001617(n) and a(A054729(n)) = -1 for all n>=1. - Gheorghe Coserea, May 22 2016

A091404 Numbers n such that genus of group Gamma_0(n) is 2.

Original entry on oeis.org

22, 23, 26, 28, 29, 31, 37, 50

Offset: 1

N. J. A. Sloane, Mar 02 2004

I assume it is known that there are no further terms? A reference for this would be nice.

B. Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 103.
G. Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see Prop. 1.40 and 1.43.

Cf. A001617, A001615, A000089, A000086, A001616, A091401, A091403.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a86[n_] := a86[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, a86[p^e]], {pe, FactorInteger[n]}];
a1615[n_] := n Sum[MoebiusMu[d]^2/d, {d, Divisors[n]}];
a1616[n_] := Sum[EulerPhi[GCD[d, n/d]], {d, Divisors[n]}];
a1617[n_] := 1 + a1615[n]/12 - a89[n]/4 - a86[n]/3 - a1616[n]/2;
Position[Array[a1617, 100], 2] // Flatten (* Jean-François Alcover, Oct 19 2018 *)

Numbers n such that A001617(n) = 2.

A295819 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 = n.

Original entry on oeis.org

0, 2, 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

Offset: 0

Seiichi Manyama, Nov 28 2017

nonn

			a(1) = 2;
(1,0) = 1 and 1^2 + 0^2 =  1.
(0,1) = 1 and 0^2 + 1^2 =  1.
a(2) = 1;
(1,1) = 1 and 1^2 + 1^2 =  2. ->  1^2 +  1^2 == 1^2 + 1 == 0 mod  2.
a(5) = 2;
(2,1) = 1 and 2^2 + 1^2 =  5. ->  2^2 +  1^2 == 2^2 + 1 == 0 mod  5.
(1,2) = 1 and 1^2 + 2^2 =  5. ->  3^2 +  6^2 == 3^2 + 1 == 0 mod  5.
a(10) = 2;
(3,1) = 1 and 3^2 + 1^2 = 10. ->  3^2 +  1^2 == 3^2 + 1 == 0 mod 10.
(1,3) = 1 and 1^2 + 3^2 = 10. ->  7^2 + 21^2 == 7^2 + 1 == 0 mod 10.
a(13) = 2;
(3,2) = 1 and 3^2 + 2^2 = 13. -> 21^2 + 14^2 == 8^2 + 1 == 0 mod 13.
(2,3) = 1 and 2^2 + 3^2 = 13. -> 18^2 + 27^2 == 5^2 + 1 == 0 mod 13.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006278.
Similar sequences: A000010, A000925, A295820, A295848, A295976.
A000089 is essentially the same sequence.

a[n_] := Sum[j = Sqrt[n - i^2] // Floor; Boole[GCD[i, j] == 1 && i^2 + j^2 == n], {i, 0, Sqrt[n]}];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Jul 05 2018, after Andrew Howroyd *)

a(n) = {sum(i=0, sqrtint(n), my(j=sqrtint(n-i^2)); gcd(i,j)==1 && i^2+j^2==n)} \\ Andrew Howroyd, Dec 12 2017

a(n) = A000089(n) for n >= 2.

a(A006278(n)) = 2^n for n >= 1.

A304651 Number of coprime pairs (x,y) with x^2 + y^2 <= n.

Original entry on oeis.org

0, 4, 8, 8, 8, 16, 16, 16, 16, 16, 24, 24, 24, 32, 32, 32, 32, 40, 40, 40, 40, 40, 40, 40, 40, 48, 56, 56, 56, 64, 64, 64, 64, 64, 72, 72, 72, 80, 80, 80, 80, 88, 88, 88, 88, 88, 88, 88, 88, 88, 96, 96, 96, 104, 104, 104, 104, 104, 112, 112, 112, 120, 120, 120, 120

Offset: 0

Seiichi Manyama, May 26 2018

nonn

			a(2) = 8 counts (x,y) = (-1,-1), (-1,0), (-1,1), (0,-1), (0,1), (1,-1), (1,0) and (1,1).

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000089, A175341, A305117.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a[n_] := a[n] = If[n == 0, 0, a[n-1] + 4 a89[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 02 2023 *)

a(n) = a(n-1) + 4*A000089(n) for n > 0.

A305117 a(n) = A304651(n)/4.

Original entry on oeis.org

0, 1, 2, 2, 2, 4, 4, 4, 4, 4, 6, 6, 6, 8, 8, 8, 8, 10, 10, 10, 10, 10, 10, 10, 10, 12, 14, 14, 14, 16, 16, 16, 16, 16, 18, 18, 18, 20, 20, 20, 20, 22, 22, 22, 22, 22, 22, 22, 22, 22, 24, 24, 24, 26, 26, 26, 26, 26, 28, 28, 28, 30, 30, 30, 30, 34, 34, 34, 34, 34, 34

Offset: 0

Seiichi Manyama, May 26 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A000089, A176562, A304651.

a89[n_] := a89[n] = Product[{p, e} = pe; Which[p < 3 && e == 1, 1, p == 2 && e > 1, 0, Mod[p, 4] == 1, 2, Mod[p, 4] == 3, 0, True, a89[p^e]], {pe, FactorInteger[n]}];
a[n_] := a[n] = If[n == 0, 0, a[n-1] + a89[n]];
Table[a[n], {n, 0, 100}] (* Jean-François Alcover, Aug 02 2023 *)

a(n) = a(n-1) + A000089(n) for n > 0.

A350872 Number of coincidence site lattices of index n in square lattice.

Original entry on oeis.org

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

Offset: 1

Andrey Zabolotskiy, Jan 20 2022

A coincidence site lattice (CSL), or coincidence sublattice, is a full-rank sublattice arising as an intersection of the parent lattice with its copy rotated around the origin. It is necessarily primitive.
A primitive sublattice of the square lattice is a CSL if it is square (i. e., similar to the parent lattice) and has odd index.
In this sequence, any two CSLs differing by any isometry are counted as distinct.
a(n) is also the number of ordered pairs of coprime integers (p, q) with p >= 0 and q > 0 such that p^2 + q^2 = n^2.

			a(5) = 2 index-5 CSLs have bases (2, 1), (-1, 2) and (1, 2), (-2, 1).

Amiram Eldar, Table of n, a(n) for n = 1..10000
Michael Baake and Peter Zeiner, Geometric enumeration problems for lattices and embedded Z-modules, arXiv:1709.07317 [math.MG], 2017; in: Aperiodic Order, vol. 2: Crystallography and Almost Periodicity, eds. M. Baake and U. Grimm, Cambridge University Press, Cambridge (2017), pp. 73-172.
Index entries for sequences related to square lattice.
Index entries for sequences related to sublattices.

Cf. A031358 (nonzero quadrisection), A004613 (positions of nonzero terms), A024362, A154269, A338690, A271102.
Cf. enumeration of wider classes of sublattices of Z^2: A000203 (all sublattices), A350871 (all well-rounded sublattices), A002654 (all square sublattices), A001615 (all primitive sublattices), A000089 (all primitive square sublattices).
Cf. enumeration of CSLs in other lattices: A331140 (Z^4), A331139 (D_4), A331142 (A_4).

csl[1] = 1;
csl[n_] := With[{f = First@Transpose@FactorInteger@n}, If[Union@Mod[f, 4] == {1}, 2^Length@f, 0]];
Array[csl, 87]

a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 1]%4 == 1, 2, 0));} \\ Amiram Eldar, Oct 23 2023

Multiplicative with a(p^e) = 2 if p == 1 (mod 4), otherwise 0.
a(4*n+1) = A031358(n), other terms are 0.
a(n) = 2 * A024362(n) for n > 1.
Dirichlet convolution of A000089 and A154269.
Dirichlet convolution of A338690 and A271102.
From Amiram Eldar, Oct 23 2023: (Start)
Dirichlet g.f.: Product_{primes p == 1 (mod 4)} (1 + 1/p^s)/(1 - 1/p^s).
Sum_{k=1..n} a(k) = (1/Pi) * n + O(sqrt(n)*log(n)).
(both from Baake and Zeiner, 2017) (End)

A000095 Number of fixed points of GAMMA_0 (n) of type i.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

			G.f. = x + 2*x^2 + 2*x^5 + 4*x^10 + 2*x^13 + 2*x^17 + 2*x^25 + 4*x^26 + 2*x^29 + ...

Bruno Schoeneberg, Elliptic Modular Functions, Springer-Verlag, NY, 1974, p. 101.
Goro Shimura, Introduction to the Arithmetic Theory of Automorphic Functions, Princeton, 1971, see p. 25, Eq. (2).

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A027748, A060294, A124010, A000089.

a000095 n = product $ zipWith f (a027748_row n) (a124010_row n) where
   f 2 e = if e == 1 then 2 else 0
   f p _ = if p `mod` 4 == 1 then 2 else 0
-- Reinhard Zumkeller, Mar 24 2012

A000095 := proc(n) local b,d: if irem(n,4) = 0 then RETURN(0); else b := 1; for d from 2 to n do if irem(n,d) = 0 and isprime(d) then b := b*(1+legendre(-1,d)); fi; od; RETURN(b); fi: end;

A000095[ 1 ] = 1; A000095[ n_Integer ] := If[ Mod[ n, 4 ]==0, 0, Fold[ #1*(1+JacobiSymbol[ -1, #2 ])&, If[ EvenQ[ n ], 2, 1 ], Select[ First[ Transpose[ FactorInteger[ n ] ] ], OddQ ] ] ]
a[ n_] := If[ n < 1, 0, Times @@ (Which[# == 1, 1, # == 2, 2 Boole[#2 == 1], Mod[#, 4] == 1, 2, True, 0] & @@@ FactorInteger[n])]; (* Michael Somos, Nov 15 2015 *)

{a(n) = my(t); if( n<=1 || n%4==0, n==1, t=1; fordiv(n, d, if( isprime(d), t *= (1 + kronecker(-1, d)))); t)}; /* Michael Somos, Jul 15 2004 */

A000095(n)=n%3 && n%4 && n%7 && n%11 && return(prod(k=1,#n=factor(n)[,1],1+kronecker(-1,n[k]))) /* the n%4 is needed, the others only reduce execution time by 34% */ \\ M. F. Hasler, Mar 24 2012

from sympy import primefactors
def A000095(n): return 0 if n%4==0 or (f:=primefactors(n)) and any(p%4==3 for p in f) else 2**len(f) # David Radcliffe, Aug 20 2025

a(n) is multiplicative with a(2) = 2, a(2^e) = 0 if e>1, a(p^e) = 2 if p == 1 mod 4 and a(p^e) = 0 if p == 3 mod 4. - Michael Somos, Jul 15 2004

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2/Pi = 0.636619... (A060294). - Amiram Eldar, Oct 15 2022

Values a(1)-a(10^4) double checked by M. F. Hasler, Mar 24 2012

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

A157224 Number of primitive inequivalent (up to Pi/2 rotation) nonsquare sublattices of square lattice of index n.

Original entry on oeis.org

0, 1, 2, 3, 2, 6, 4, 6, 6, 8, 6, 12, 6, 12, 12, 12, 8, 18, 10, 18, 16, 18, 12, 24, 14, 20, 18, 24, 14, 36, 16, 24, 24, 26, 24, 36, 18, 30, 28, 36, 20, 48, 22, 36, 36, 36, 24, 48, 28, 44, 36, 42, 26, 54, 36, 48, 40, 44, 30, 72, 30, 48, 48, 48, 40, 72, 34, 54

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. A000089 (primitive square sublattices), A002654 (all square sublattices), A145392 (all sublattices), A001615, A304182.

a(n) = (A001615(n) - A000089(n))/2. - Andrey Zabolotskiy, May 09 2018

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

cp's OEIS Frontend

A091403 Numbers n such that genus of group Gamma_0(n) is 1.

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A054728 a(n) is the smallest level N such that genus of modular curve X_0(N) is n (or -1 if no such curve exists).

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A091404 Numbers n such that genus of group Gamma_0(n) is 2.

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A295819 Number of nonnegative solutions to (x,y) = 1 and x^2 + y^2 = n.

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A304651 Number of coprime pairs (x,y) with x^2 + y^2 <= n.

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A305117 a(n) = A304651(n)/4.

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A350872 Number of coincidence site lattices of index n in square lattice.

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A000095 Number of fixed points of GAMMA_0 (n) of type i.

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