A054522 -id:A054522 - cp's OEIS frontend

A054521 Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 1, 0, 1, 1, 0, 1, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 0, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 0, 1, 0

Offset: 1

N. J. A. Sloane, Apr 09 2000

Row sums = phi(n), A000010: (1, 1, 2, 2, 4, 2, 6, ...). - Gary W. Adamson, May 20 2007
Characteristic function of A169581: a(A169581(n)) = 1; a(A169582(n)) = 0. - Reinhard Zumkeller, Dec 02 2009
The function T(n,k) = T(k,n) is defined for k > n but only the values for 1 <= k <= n as a triangular array are listed here.
T(n,k) = |K(n-k|k)| where K(i|j) is the Kronecker symbol. - Peter Luschny, Aug 05 2012
Twice the sum over the antidiagonals, starting with entry T(n-1,1), for n >= 3, is the same as the row n sum (i.e., phi(n): 2*Sum_{k=1..floor(n/2)} T(n-k,k) = phi(n), n >= 3). - Wolfdieter Lang, Apr 26 2013
The number of zeros in the n-th row of the triangle is cototient(n) = A051953(n). - Omar E. Pol, Apr 21 2017
This triangle is the j = 1 sub-triangle of A349221(n,k) = Sum_{j>=1} [k|binomial(n-1,k-1) AND gcd(n,k) = j], n >= 1, 1 <= k <= n, where [] is the Iverson bracket. - Richard L. Ollerton, Dec 14 2021

			The triangle T(n,k) begins:
  n\k  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 ...
   1:  1
   2:  1  0
   3:  1  1  0
   4:  1  0  1  0
   5:  1  1  1  1  0
   6:  1  0  0  0  1  0
   7:  1  1  1  1  1  1  0
   8:  1  0  1  0  1  0  1  0
   9:  1  1  0  1  1  0  1  1  0
  10:  1  0  1  0  0  0  1  0  1  0
  11:  1  1  1  1  1  1  1  1  1  1  0
  12:  1  0  0  0  1  0  1  0  0  0  1  0
  13:  1  1  1  1  1  1  1  1  1  1  1  1  0
  14:  1  0  1  0  1  0  0  0  1  0  1  0  1  0
  15:  1  1  0  1  0  0  1  1  0  0  1  0  1  1  0
  ... (Reformatted by _Wolfdieter Lang_, Apr 26 2013)
Sums over antidiagonals: n = 3: 2*T(2,1) = 2 = T(3,1) + T(3,2) = phi(3). n = 4: 2*(T(3,1) + T(2,2)) = 2 = phi(4), etc. - _Wolfdieter Lang_, Apr 26 2013

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Jakub Jaroslaw Ciaston, A054531 vs A164306 (plot shows these ones)
Index entries for characteristic functions

Cf. A051731, A054522, A215200.
Cf. A050873, A063524.
Cf. A349221.

a054521 n k = a054521_tabl !! (n-1) !! (k-1)
a054521_row n = a054521_tabl !! (n-1)
a054521_tabl = map (map a063524) a050873_tabl
a054521_list = concat a054521_tabl
-- Reinhard Zumkeller, Sep 03 2015

A054521_row := n -> seq(abs(numtheory[jacobi](n-k,k)),k=1..n);
for n from 1 to 13 do A054521_row(n) od; # Peter Luschny, Aug 05 2012

T[ n_, k_] := Boole[ n>0 && k>0 && GCD[ n, k] == 1] (* Michael Somos, Jul 17 2011 *)
T[ n_, k_] := If[ n<1 || k<1, 0, If[ k>n, T[ k, n], If[ k==1, 1, If[ n>k, T[ k, Mod[ n, k, 1]], 0]]]] (* Michael Somos, Jul 17 2011 *)

{T(n, k) = n>0 && k>0 && gcd(n, k)==1} /* Michael Somos, Jul 17 2011 */

def A054521_row(n): return [abs(kronecker_symbol(n-k,k)) for k in (1..n)]
for n in (1..13): print(A054521_row(n)) # Peter Luschny, Aug 05 2012

T(n,k) = A063524(A050873(n,k)). - Reinhard Zumkeller, Dec 02 2009, corrected Sep 03 2015

T(n,k) = A054431(n,k) = A054431(k,n). - R. J. Mathar, Jul 21 2016

A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

Original entry on oeis.org

1, 1, 1, 1, 3, 2, 1, 9, 8, 6, 1, 25, 20, 30, 24, 20, 1, 75, 80, 180, 144, 240, 1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420, 1, 763, 1232, 5460, 1344, 10640, 5760, 5040, 0, 4032, 0, 3360, 0, 0, 2688, 1, 2619, 5768, 30996, 3024, 83160, 25920, 45360, 40320, 27216, 0, 30240, 0, 25920, 24192, 0, 0, 0, 0, 18144

Offset: 1

Roger Cuculière, Oct 29 2000

Every row for n >= 7 contains zeros. Landau's function quickly becomes > 2*n, and there is always a prime between n and 2*n. T(n,p) = 0 for such a prime p. - Franklin T. Adams-Watters, Oct 25 2011

			Triangle begins:
  1;
  1,   1;
  1,   3,   2;
  1,   9,   8,   6;
  1,  25,  20,  30,  24,   20;
  1,  75,  80, 180, 144,  240;
  1, 231, 350, 840, 504, 1470, 720, 0, 0, 504, 0, 420;
  ...

Herbert S. Wilf, "The asymptotics of e^P(z) and the number of elements of each order in S_n." Bull. Amer. Math. Soc., 15.2 (1986), 225-232.

Alois P. Heinz, Rows n = 1..30, flattened
FindStat - Combinatorial Statistic Finder, The order of a permutation.
Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages). - _N. J. A. Sloane_, Mar 27 2015

Cf. A000793, also A054522 (for cyclic group), A057740 (alternating group), A057741 (dihedral group).
Rows sums give A000142, last elements of rows give A074859, columns k=2, 3, 5, 7, 11 give A001189, A001471, A059593, A153760, A153761. - Alois P. Heinz, Feb 16 2013
Main diagonal gives A074351.
Cf. A222029.

Magma
```
{* Order(g) : g in Sym(6) *};
```

with(group):
for n from 1 do
    f := [seq(0,i=1..n!)] ;
    mknown := 0 ;
    # loop through the permutations of n
    Sn := combinat[permute](n) ;
    for per in Sn do
        # write this permutation in cycle notation
        gen := convert(per,disjcyc) ;
        # compute the list of lengths of the cycles, then the lcm of these
        cty := [seq(nops(op(i,gen)),i=1..nops(gen))] ;
        if cty <> [] then
            lcty := lcm(op(cty)) ;
        else
            lcty := 1 ;
        end if;
        f := subsop(lcty = op(lcty,f)+1,f) ;
        mknown := max(mknown,lcty) ;
    end do:
    ff := add(el,el=f) ;
    print(seq(f[i],i=1..mknown)) ;
end do: # R. J. Mathar, May 26 2014
# second Maple program:
b:= proc(n, g) option remember; `if`(n=0, x^g, add((j-1)!
      *b(n-j, ilcm(g, j))*binomial(n-1, j-1), j=1..n))
    end:
T:= n-> (p-> seq(coeff(p, x, i), i=1..degree(p)))(b(n, 1)):
seq(T(n), n=1..12);  # Alois P. Heinz, Jul 11 2017

<Jean-François Alcover, Aug 31 2016 *)
b[n_, g_] := b[n, g] = If[n == 0, x^g, Sum[(j-1)!*b[n-j, LCM[g, j]]* Binomial[n-1, j-1], {j, 1, n}]];
T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 1, Exponent[p, x]}]][ b[n, 1]];
Array[T, 12] // Flatten (* Jean-François Alcover, May 03 2019, after Alois P. Heinz *)

T(n,k)={n!*polcoeff(sumdiv(k, i, moebius(k/i)*exp(sumdiv(i, j, x^j/j) + O(x*x^n))), n)} \\ Andrew Howroyd, Jul 02 2018

Sum_{k=1..A000793(n)} k*T(n,k) = A060014(n); A000793 = Landau's function.

More terms from N. J. A. Sloane, Nov 01 2000

A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

Original entry on oeis.org

1, 1, 1, 0, 2, 1, 3, 8, 1, 15, 20, 0, 24, 1, 45, 80, 90, 144, 1, 105, 350, 630, 504, 210, 720, 1, 315, 1232, 3780, 1344, 5040, 5760, 0, 0, 0, 0, 0, 0, 0, 2688, 1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192

Offset: 1

Roger Cuculière, Oct 29 2000

			Triangle begins:
  1;
  1;
  1,    0,    2;
  1,    3,    8;
  1,   15,   20,     0,   24;
  1,   45,   80,    90,  144;
  1,  105,  350,   630,  504,   210,   720;
  1,  315, 1232,  3780, 1344,  5040,  5760, 0,     0,    0, 0,     0, 0, 0,  2688;
  1, 1323, 5768, 18900, 3024, 37800, 25920, 0, 40320, 9072, 0, 15120, 0, 0, 24192;
...

J. H. Conway, R. T. Curtis, S. P. Norton, R. A. Parker and R. A. Wilson, ATLAS of Finite Groups. Oxford Univ. Press, 1985 [for best online version see https://oeis.org/wiki/Welcome#Links_to_Other_Sites].

Koda, Tatsuhiko; Sato, Masaki; Takegahara, Yugen; 2-adic properties for the numbers of involutions in the alternating groups, J. Algebra Appl. 14 (2015), no. 4, 1550052 (21 pages).
Index entries for sequences related to groups

Cf. A057731 (S_n), A054522, A057741, A051593, A000793.
Column 2, 3, 4 are A048099, A001471, A051695.
See also A061129, A061130, A061131, A061132, A061133, A061134, A061135, A061136, A061137, A061138, A061139, A061140.

Magma
```
{* Order(g) : g in Alt(6) *};
```

row[n_] := (orders = PermutationOrder /@ GroupElements[AlternatingGroup[n] ]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 9}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

More terms from N. J. A. Sloane, Nov 01 2000

Missing zero in the row for A_9 inserted by N. J. A. Sloane, Mar 27 2015

A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

Original entry on oeis.org

1, 2, 2, 3, 0, 6, 4, 4, 0, 8, 5, 0, 0, 0, 20, 6, 6, 12, 0, 0, 12, 7, 0, 0, 0, 0, 0, 42, 8, 8, 0, 16, 0, 0, 0, 32, 9, 0, 18, 0, 0, 0, 0, 0, 54, 10, 10, 0, 0, 40, 0, 0, 0, 0, 40

Offset: 1

Gary W. Adamson, Jan 15 2007

Mobius transform of A127481.

			First few rows of the triangle are:
1;
2, 2;
3, 0, 6;
4, 4, 0, 8;
5, 0, 0, 0, 20;
6, 6, 12, 0, 0, 12;
7, 0, 0, 0, 0, 0, 42;
8, 8, 0, 16, 0, 0, 0, 32;
...

Cf. A054522, A127093, A062952, A002618.

Cf. A127093, A127481, A054525.

A127466 := proc(n,k)
    add( A054525(n,j)*A127481(j,k),j=k..n) ;
end proc: # R. J. Mathar, Sep 06 2013

Sum_{k=1..n} T(n,k) = n^2.

T(n,n) = A002618(n) = n*phi(n).

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, Aug 23 2007

A130212 T(k, n) = sum_(1 <= j <= k) [j | k] j mu(k / j) floor(n / k), triangle read by rows.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 4, 2, 2, 2, 5, 2, 2, 2, 4, 6, 3, 4, 2, 4, 2, 7, 3, 4, 2, 4, 2, 6, 8, 4, 4, 4, 4, 2, 6, 4, 9, 4, 6, 4, 4, 2, 6, 4, 6, 10, 5, 6, 4, 8, 2, 6, 4, 6, 4, 11, 5, 6, 4, 8, 2, 6, 4, 6, 4, 10, 12, 6, 8, 6, 8, 4, 6, 4, 6, 4, 10, 4, 13, 6, 8, 6, 8, 4, 6, 4, 6, 4, 10, 4, 12

Offset: 1

Gary W. Adamson, May 17 2007

			First few rows of the triangle are:
1;
2, 1;
3, 1, 2;
4, 2, 2, 2;
5, 2, 2, 2, 4;
6, 3, 4, 2, 4, 2;
7, 3, 4, 2, 4, 2, 6;
8, 4, 4, 4, 4, 2, 6, 4;
9, 4, 6, 4, 4, 2, 6, 4, 6;
10, 5, 6, 4, 8, 2, 6, 4, 6, 4;
...

Cf. A000010, A130211 (product with swapped matrices), A054522, A000217 (row sums).

with(numtheory): A130212 := (n, k) -> add(j*mobius(k / j)*iquo(n, k), j = divisors(k)); # Peter Luschny, Oct 28 2010

A[n_, k_] := Sum[j MoebiusMu[k/j] Floor[n/k], {j, Divisors[k]}];
Table[A[n, k], {n, 1, 13}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jun 14 2019 *)

A000012 * A054522 as infinite lower triangular matrices (previous name).

T(n,n) = A000010(n).

Name replaced by new formula by Peter Luschny, Oct 28 2010

T(6,1) corrected by R. J. Mathar, Aug 06 2016

A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 2, 1, 5, 0, 2, 1, 5, 0, 0, 4, 1, 7, 2, 0, 0, 2, 1, 7, 0, 0, 0, 0, 6, 1, 9, 0, 2, 0, 0, 0, 4, 1, 9, 2, 0, 0, 0, 0, 0, 6, 1, 11, 0, 0, 4, 0, 0, 0, 0, 4, 1, 11, 0, 0, 0, 0, 0, 0, 0, 0, 10, 1, 13, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 13, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 12, 1, 15, 0, 0, 0, 0, 6, 0, 0

Offset: 1

Roger Cuculière, Oct 29 2000

Note that D_2 equals the cyclic group of order 2.

			1,1;
1,3;
1,3,2;
1,5,0,2;
1,5,0,0,4; ...

Cf. A057731, A054522, A057740.

t[n_, k_] /; k != 2 && ! Divisible[n, k] = 0; t[n_, k_] /; k != 2 && Divisible[n, k] := EulerPhi[k]; t[n_, 2] := n + 1 - Mod[n, 2]; Flatten[Table[t[n, k], {n, 1, 14}, {k, 1, If[n == 1, 2, n]}]] (* Jean-François Alcover, Jun 19 2012, from formula *)
row[n_] := (orders = PermutationOrder /@ GroupElements[DihedralGroup[n]]; Table[Count[orders, k], {k, 1, Max[orders]}]); Table[row[n], {n, 1, 14}] // Flatten (* Jean-François Alcover, Aug 31 2016 *)

If k<>2 and k does not divide n, this number is 0; if k<>2 and k divides n, this number is phi(k), where phi is the Euler totient function; if k=2, this number is n for odd n and n+1 for even n.

More terms from James Sellers, Oct 30 2000

A127481 Triangle T(n,k) read by rows: T(n,k) = sum_{l=k..n, l|n, k|l} l*phi(k).

Original entry on oeis.org

1, 3, 2, 4, 0, 6, 7, 6, 0, 8, 6, 0, 0, 0, 20, 12, 8, 18, 0, 0, 12, 8, 0, 0, 0, 0, 0, 42, 15, 14, 0, 24, 0, 0, 0, 32, 13, 0, 24, 0, 0, 0, 0, 0, 54, 18, 12, 0, 0, 60, 0, 0, 0, 0, 40, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 110, 28, 24, 42, 32, 0, 36, 0, 0, 0, 0, 0, 48, 14, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Gary W. Adamson, Jan 15 2007

			First few rows of the triangle are:
1;
3, 2;
4, 0, 6;
7, 6, 0, 8;
6, 0, 0, 0, 20,
12, 8, 18, 0, 0, 12;
8, 0, 0, 0, 0, 0, 42;
15, 14, 0, 24, 0, 0, 0, 32;
...

Cf. A054522, A127093, A001157 (row sums), A002618, A127466.

A127481 := proc(n,k)
    a :=0 ;
    for l from k to n do
        if modp(n,l) =0 and modp(l,k) =0 then
            a := a+l*numtheory[phi](k) ;
        end if;
    end do:
    a ;
end proc: # R. J. Mathar, Sep 06 2013

T(n,1) = A000203(n).
T(n,n) = A002618(n).
T(n,k) =sum_{l=k..n} A127093(n,l) * A054522(l,k), the matrix product of the infinite lower triangular matrices.

A252911 Irregular triangular array read by rows: T(n,k) is the number of elements in the multiplicative group of integers modulo n that have order k, n>=1, 1<=k<=A002322(n).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 0, 2, 1, 1, 1, 1, 2, 0, 0, 2, 1, 3, 1, 1, 2, 0, 0, 2, 1, 1, 0, 2, 1, 1, 0, 0, 4, 0, 0, 0, 0, 4, 1, 3, 1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 2, 0, 0, 2, 1, 3, 0, 4, 1, 3, 0, 4, 1, 1, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 0, 0, 2, 1, 1, 2, 0, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 6, 1, 3, 0, 4

Offset: 1

Geoffrey Critzer, Dec 24 2014

Row sums are A000010.

Column 2 = A155828(n) = A060594(n) - 1.

			1;
1;
1, 1;
1, 1;
1, 1, 0, 2;
1, 1;
1, 1, 2, 0, 0, 2;
1, 3;
1, 1, 2, 0, 0, 2;
1, 1, 0, 2;
1, 1, 0, 0, 4, 0, 0, 0, 0, 4;
1, 3;
1, 1, 2, 2, 0, 2, 0, 0, 0, 0, 0, 4;
1, 1, 2, 0, 0, 2;
1, 3, 0, 4;
T(15,2)=3 because the elements 4, 11, and 14 have order 2 in the modulo multiplication group (Z/15Z)*. We observe that 4^2, 11^2, and 14^2 are congruent to 1 mod 15.

Alois P. Heinz, Rows n = 1..250, flattened
Eric Weisstein's World of Mathematics, Modulo Multiplication Group.

Cf. A000010, A002322, A054522, A060594, A155828.

with(numtheory):
T:= n-> `if`(n=1, 1, (p-> seq(coeff(p, x, j), j=1..degree(p)))(
         add(`if`(igcd(n, i)>1, 0, x^order(i, n)), i=1..n-1))):
seq(T(n), n=1..30);  # Alois P. Heinz, Dec 30 2014

Table[Table[
   Count[Table[
     MultiplicativeOrder[a, n], {a,
      Select[Range[n], GCD[#, n] == 1 &]}], k], {k, 1,
    CarmichaelLambda[n]}], {n, 1, 20}] // Grid

A345628 Irregular triangle T(n,k) read by rows of the number of elements of order k in the dicyclic group Dic(n) for n>=2.

Original entry on oeis.org

1, 1, 0, 6, 1, 1, 2, 6, 0, 2, 1, 1, 0, 10, 0, 0, 0, 4, 1, 1, 0, 10, 4, 0, 0, 0, 0, 4, 1, 1, 2, 14, 0, 2, 0, 0, 0, 0, 0, 4, 1, 1, 0, 14, 0, 0, 6, 0, 0, 0, 0, 0, 0, 6, 1, 1, 0, 18, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 8, 1, 1, 2, 18, 0, 2, 0, 0, 6, 0, 0, 0, 0, 0, 0

Offset: 2

Sean A. Irvine, Jun 22 2021

Dic(1) is omitted since it is degenerate.

Row n has 2*n entries (k=1..2*n).

			Triangle begins:
  1, 1, 0,  6;
  1, 1, 2,  6, 0, 2;
  1, 1, 0,  1, 0, 0, 0, 4;
  1, 1, 0, 10, 4, 0, 0, 0, 0, 4;
  ...

Sean A. Irvine, Rows n=2..100 flattened
Sean A. Irvine, Java program (github)
Wikipedia, Dicyclic group

Cf. A054522, A057731.

A127472 Triangle T(n,k) = Sum_{j=k..n, j|n, k|j} phi(j) read by rows, 1<=k<=n.

Original entry on oeis.org

1, 2, 1, 3, 0, 2, 4, 3, 0, 2, 5, 0, 0, 0, 4, 6, 3, 4, 0, 0, 2, 7, 0, 0, 0, 0, 0, 6, 8, 7, 0, 6, 0, 0, 0, 4, 9, 0, 8, 0, 0, 0, 0, 0, 6, 10, 5, 0, 0, 8, 0, 0, 0, 0, 4, 11, 0, 0, 0, 0, 0, 0, 0, 0, 0, 10, 12, 9, 8, 6, 0, 6, 0, 0, 0, 0, 0, 4, 13

Offset: 1

Gary W. Adamson, Jan 15 2007

Defined by the matrix product A054522 * A051731.

			First few rows of the triangle are;
.1;
.2, 1;
.3, 0, 2;
.4, 3, 0, 2;
.5, 0, 0, 0, 4;
.6, 3, 4, 0, 0, 2;
.7, 0, 0, 0, 0, 0, 6;
.8, 7, 0, 6, 0, 0, 0, 4;
....

Cf. A054522, A051731, A062949 (row sums), A000010 (diagonal n=k), A127471 (swapped matrix product).

A127472 := proc(n,k)
        a := 0 ;
        for j from k to n do
                if (n mod j = 0 ) and (j mod k =0 ) then
                        a := a+numtheory[phi](j) ;
                end if;
        end do;
        a ;
end proc:
seq(seq(A127472(n,k),k=1..n),n=1..14) ; # R. J. Mathar, Nov 11 2011

T(n,k) = Sum_{j=k..n} A054522(n,j) * A051731(j,k), 1<=k<=n.

cp's OEIS Frontend

A054521 Triangle read by rows: T(n,k) = 1 if gcd(n, k) = 1, T(n,k) = 0 otherwise (n >= 1, 1 <= k <= n).

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A057731 Irregular triangle read by rows: T(n,k) = number of elements of order k in symmetric group S_n, for n >= 1, 1 <= k <= g(n), where g(n) = A000793(n) is Landau's function.

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PARI

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A057740 Irregular triangle read by rows: T(n,k) is the number of elements of alternating group A_n having order k, for n >= 1, 1 <= k <= A051593(n).

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Magma

Mathematica

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A127466 Triangle read by rows: A054525 * A127481 as infinite lower triangular matrices.

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Maple

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A130212 T(k, n) = sum_(1 <= j <= k) [j | k] j mu(k / j) floor(n / k), triangle read by rows.

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Maple

Mathematica

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A057741 Table T(n,k) giving number of elements of order k in dihedral group D_{2n} of order 2n, n >= 1, 1<=k<=g(n), where g(n) = 2 if n=1 else g(n) = n.

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Mathematica

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A127481 Triangle T(n,k) read by rows: T(n,k) = sum_{l=k..n, l|n, k|l} l*phi(k).

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