Alexander Gruber - cp's OEIS frontend

A239202 Multiplicative order of phi(n) modulo n when gcd(phi(n),n)=1.

1, 1, 2, 2, 2, 2, 2, 4, 2, 2, 2, 2, 2, 10, 6, 2, 2, 2, 2, 8, 2, 2, 2, 12, 2, 22, 2, 2, 15, 2, 2, 4, 28, 2, 12, 36, 2, 2, 2, 2, 2, 2, 44, 48, 20, 2, 2, 18, 2, 2, 46, 6, 28, 2, 2, 2, 52, 22, 2, 2, 2, 58, 2, 2, 18, 80, 2, 2, 2, 2, 45, 2, 70, 28, 6, 48, 2, 2, 2

Offset: 1

Alexander Gruber, Mar 12 2014

nonn

			For n = 8: the 8th entry of A003277 is 15, and phi(15) = 8 has multiplicative order 4 modulo 15, so a(8) = 4.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Indexed by A003277.

MultiplicativeOrder[EulerPhi[#], #] & /@ Select[Range[1000], GCD[#, EulerPhi[#]] == 1 &]

lista(nn) = {for(n=1, nn, my(ephi = eulerphi(n)); if (gcd(ephi, n) == 1, print1(znorder(Mod(ephi, n)), ", ")););} \\ Michel Marcus, Feb 09 2015

A211877 Number of involutions in GL(n,4).

Original entry on oeis.org

1, 21, 673, 102273, 47663617, 110981851137, 815432848809985, 30052835284679819265, 3519512226295269640765441, 2069751512310185039905834926081, 3874510079394593253089862950754189313, 36431456010689490638771956423547489198538753

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..50

A211877 := proc(n)
    add( 4^(k*(n-k))*A022168(n,k),k=1..n) ;
end proc: # R. J. Mathar, Apr 26 2013

Sum[q^(k (n - k)) QBinomial[n, k, q], {k, 1, n}]

A211876 Primes of the form Phi(phi(k),2), the phi(k)-th cyclotomic polynomial evaluated at 2, where phi is the Euler totient function.

Original entry on oeis.org

3, 5, 11, 13, 17, 241, 257, 331, 683, 5419, 61681, 65537, 2796203, 15790321, 22366891, 4278255361, 4562284561, 77158673929, 1133836730401, 18446744069414584321, 291280009243618888211558641, 78919881726271091143763623681, 84159375948762099254554456081, 84179842077657862011867889681

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Cf. A000010, A211875, A211874.

s = Union[Table[EulerPhi[n], {n, 2000}]]; t = Union[Select[Table[ Cyclotomic[ n, 2], {n, s}], PrimeQ]]; Select[t, # < 10^30 &]

s=Set([]);
for (n=1,10^3, my(a=polcyclo(eulerphi(n),2)); if(ispseudoprime(a), s=setunion(s,[a])));
v211876=s  /* Joerg Arndt, Apr 13 2013 */

A214082 Size of conjugacy classes of GL(4,2) arranged by order of elements in that class.

Original entry on oeis.org

1, 105, 210, 112, 1120, 1260, 2520, 1344, 1680, 3360, 2880, 2880, 1344, 1344

Offset: 1

Alexander Gruber, Feb 15 2013

[ g[2] : g in ConjugacyClasses(GL(4,2)) ];

A211874 Primes of the form Phi_k(3), the k-th cyclotomic polynomial evaluated at 3.

Original entry on oeis.org

2, 7, 13, 61, 73, 547, 757, 1093, 4561, 6481, 368089, 398581, 530713, 797161, 42521761, 47763361, 2413941289, 23535794707, 282429005041, 374857981681, 144542918285300809, 150094634909578633, 13490012358249728401, 82064241848634269407

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Primes in A019321.

Cf. A138933, A072226, A153601.

Sort[Select[Cyclotomic[Range[1000], 3], PrimeQ]]

A211875 Primes of the form Phi(phi(k),3), the phi(k)-th Cyclotomic polynomial evaluated at 3, where phi is the Euler totient function.

Original entry on oeis.org

2, 7, 61, 73, 6481, 530713, 42521761, 47763361, 23535794707, 282429005041, 374857981681, 150094634909578633, 13490012358249728401, 105919308797935444986721, 1076050302914923449767311155851656076154481

Offset: 1

Alexander Gruber, Feb 12 2013

nonn

Union[Select[Cyclotomic[EulerPhi[#], x] /. x -> 3 & /@ Range[1000], PrimeQ]]

A211391 The number of divisors d of n! such that d < A000793(n) (Landau's function g(n)) and the symmetric group S_n contains no elements of order d.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 2, 2, 2, 6, 4, 15, 15, 24, 29, 33, 63, 55, 126, 117, 110, 103, 225, 212, 288, 282, 319, 428, 504, 774, 859, 943, 924, 1336, 1307, 1681, 1869, 2097, 2067, 2866, 3342, 3487, 5612, 5567, 5513, 5549, 9287, 9220, 11594, 11524, 11481, 11403, 18690

Offset: 1

Alexander Gruber, Feb 07 2013

nonn

This sequence gives the number of divisors d of |S_n| such that d < Lambda(n) (where Lambda(n) = the largest order of an element in S_n) for which S_n contains no element of order d. These divisors constitute a set of 'missing' element orders of S_n.

For computational purposes, the smallest divisor d0(n) of n! = |S_n| for which S_n has no element of order d0(n) is the smallest divisor of n! which is not the least common multiple of an integer partition of n. Thus d0(n) is given by the smallest prime power >= n+1 that is not prime (with the exception of n = 3 and 4, for which d0(n) = 6).

			For n = 7, we refer to the following table:
Symmetric Group on 7 letters.
  # of elements of order  1 ->    1
  # of elements of order  2 ->  231
  # of elements of order  3 ->  350
  # of elements of order  4 ->  840
  # of elements of order  5 ->  504
  # of elements of order  6 -> 1470
  # of elements of order  7 ->  720
  # of elements of order  8 ->    0
  # of elements of order  9 ->    0
  # of elements of order 10 ->  504
  # of elements of order 12 ->  420
  (All other divisors of 7! -> 0.)
So there are two missing element orders in S_7, whence a(7) = 2.

d0(n) is equal to A167184(n) for n >= 5.

Cf. A000793 (Landau's function g(n)), A057731, A211392.

for n in [1..25] do
D := Set(Divisors(Factorial(n)));
O := { LCM(s) : s in Partitions(n) };
L := Max(O);
N := D diff O;
#{ n : n in N | n lt L };
end for;

More terms from Alois P. Heinz, Feb 11 2013

A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.

Original entry on oeis.org

0, 0, 1, 4, 10, 24, 51, 85, 146, 254, 520, 769, 1557, 2561, 3997, 5333, 10705, 14633, 29315, 40970, 60722, 95912, 191902, 242769, 339909, 532088, 677224, 917112, 1834373, 2332596, 4665375, 5529352, 7864049, 12164824, 16422587, 19595164, 39190653, 60465758

Offset: 1

Alexander Gruber, Feb 07 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 1..1000

Cf. A000005, A009490, A027423, A211391.

b:= proc(n,i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0 or i<1, 1, b(n, i-1)+
      add(b(n-p^j, i-1), j=1..ilog[p](n)))
    end:
a:= n-> numtheory[tau](n!) -b(n, numtheory[pi](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 15 2013

b[n_, i_] := b[n, i] = Module[{p}, p = If[i<1, 1, Prime[i]]; If[n==0 || i<1, 1, b[n, i-1] + Sum[b[n-p^j, i-1], {j, 1, Floor@Log[p, n]}]]];
a[n_] := DivisorSigma[0, n!] - b[n, PrimePi[n]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Mar 24 2017, after Alois P. Heinz *)

a(n) = A000005(n!) - A009490(n).

More terms from Alois P. Heinz, Feb 11 2013

A211171 Exponent of general linear group GL(n,2).

Original entry on oeis.org

1, 6, 84, 420, 26040, 78120, 9921240, 168661080, 24624517680, 270869694480, 554470264600560, 7208113439807280, 59041657185461430480, 2538791258974841510640, 383357480105201068106640, 98522872387036674503406480, 25826982813282567927671981480160

Offset: 1

Alexander Gruber, Jan 31 2013

nonn

a(n) is the smallest integer for which x^a(n) = 1 for any x in GL(n,2).

			n = 2: GL(2,2) is isomorphic to S3 which has exponent 6 (see: A003418).
n = 3: The set of element orders of GL(3,2) is {1,2,3,4,7} so the exponent is 84.
n = 5: The set of element orders of GL(5,2) is {1,2,3,4,5, 6,7,8,12,14, 15,21,31} so the exponent is 26040 (see: A053651).

Alexander Gruber, Table of n, a(n) for n = 1..100
Gert Almkvist, Powers of a matrix with coefficients in a Boolean ring, Proc. Amer. Math. Soc. 53 (1975), 27-31. See u_n.
Eugene Karolinsky and Dmytro Seliutin, Carmichael numbers for GL(m), arXiv:2001.10315 [math.NT], 2020; where a(n) is noted as K2(n), see page 1.
MathStackExchange, Exponent of GL(n,q)

Cf. A003418, A053651.
Cf. A006951 (number of conjugacy classes in GL(n,2)).
Cf. A034268, A062383.

for n in [1..18] do
Exponent(GL(n,2));
end for;

with(numtheory):
a:= proc(n) local t; t:= 2^ilog2(n);
      `if`(tAlois P. Heinz, Feb 04 2013

f[q_, n_] := With[{p = Sort[Divisors[q]][[2]]},
  p^Ceiling[Log[p, n]] Product[Cyclotomic[k, q], {k, n}]]; f[2,#]&/@Range[100]

a(n) = 2^ceil(log(n)/log(2))*prod(k=1, n, polcyclo(k, 2)); \\ Michel Marcus, Jan 29 2020

a(n) = 2^ceiling(log_2(n)) * Product_{k=1..n} (k-th cyclotomic polynomial evaluated at 2).

a(n) = A034268(n)*A062383(n+1). - Michel Marcus, Jul 29 2022

A211168 Exponent of alternating group An.

Original entry on oeis.org

1, 1, 3, 6, 30, 60, 420, 420, 1260, 2520, 27720, 27720, 360360, 360360, 360360, 360360, 6126120, 12252240, 232792560, 232792560, 232792560, 232792560, 5354228880, 5354228880, 26771144400, 26771144400, 80313433200, 80313433200, 2329089562800, 2329089562800

Offset: 1

Alexander Gruber, Jan 31 2013

a(n) is the smallest natural number m such that g^m = 1 for any g in An.

If m <= n, a m-cycle occurs in some permutation in An if and only if m is odd or m <= n - 2. The exponent is the LCM of the m's satisfying these conditions, leading to the formula below.

			For n = 7, lcm{1,...,5,7} = 420.

Alexander Gruber, Table of n, a(n) for n = 1..2308

Even entries given by the sequence A076100, or the odd entries in the sequence A003418.

The records of this sequence are a subsequence of A002809 and A126098.

for n in [1..40] do
Exponent(AlternatingGroup(n));
end for;

for n in [1..40] do
if n mod 2 eq 0 then
L := [1..n-1];
else
L := Append([1..n-2],n);
end if;
LCM(L);
end for;

Table[If[Mod[n, 2] == 0, LCM @@ Range[n - 1],
  LCM @@ Join[Range[n - 2], {n}]], {n, 1, 100}] (* or *)
a[1] = 1; a[2] = 1; a[3] = 3; a[n_] := a[n] =
  If[Mod[n, 2] == 0, LCM[a[n - 1], n - 2], LCM[a[n - 2], n - 3, n]]; Table[a[n], {n, 1, 40}]

a(n)=lcm(if(n%2,concat([2..n-2],n),[2..n-1])) \\ Charles R Greathouse IV, Mar 02 2014

Explicit:
a(n) = lcm{1, ..., n-1} if n is even.
     = lcm{1, ..., n-2, n} if n is odd.
Recursive:
Let a(1) = a(2) = 1 and a(3) = 3.  Then
a(n) = lcm{a(n-1), n-2} if n is even.
     = lcm{a(n-2), n-3, n} if n is odd.
a(n) = A003418(n)/(1 + [n in A228693]) for n > 1. - Charlie Neder, Apr 25 2019

cp's OEIS Frontend

User: Alexander Gruber

A239202 Multiplicative order of phi(n) modulo n when gcd(phi(n),n)=1.

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A211877 Number of involutions in GL(n,4).

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A211876 Primes of the form Phi(phi(k),2), the phi(k)-th cyclotomic polynomial evaluated at 2, where phi is the Euler totient function.

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A214082 Size of conjugacy classes of GL(4,2) arranged by order of elements in that class.

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Magma

A211874 Primes of the form Phi_k(3), the k-th cyclotomic polynomial evaluated at 3.

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A211875 Primes of the form Phi(phi(k),3), the phi(k)-th Cyclotomic polynomial evaluated at 3, where phi is the Euler totient function.

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A211391 The number of divisors d of n! such that d < A000793(n) (Landau's function g(n)) and the symmetric group S_n contains no elements of order d.

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A211392 The number of divisors d of n! such that the symmetric group on n letters contains no elements of order d.

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Maple

Mathematica

Formula

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A211171 Exponent of general linear group GL(n,2).

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Magma

Maple

Mathematica