A060793 -id:A060793 - cp's OEIS frontend

A352287 Numbers k such that, for every prime p dividing k, k has a nontrivial divisor which is congruent to 1 (mod p).

1, 12, 24, 30, 36, 48, 56, 60, 72, 80, 90, 96, 105, 108, 112, 120, 132, 144, 150, 160, 168, 180, 192, 210, 216, 224, 240, 252, 264, 270, 280, 288, 300, 306, 315, 320, 324, 336, 351, 360, 380, 384, 392, 396, 400, 420, 432, 448, 450, 480, 495, 504, 520, 525, 528, 540, 546, 552, 560, 576, 600

Offset: 1

David Speyer, Mar 10 2022

nonn

When considering whether an integer k is the order of a finite simple group, the first thing one checks is whether the number of p-Sylow subgroups is forced to be 1 for some p dividing k. This occurs if the only divisor of k which is 1 (mod p) is 1 itself. This sequence consists of the numbers that survive this test.

			105 is in the sequence, since it is divisible by 7 which is 1 (mod 3), 21 which is 1 (mod 5), and 15 which is 1 (mod 7).

Peter Luschny, Table of n, a(n) for n = 1..10000
Mariano Suárez-Álvarez, On the density of the orders excluded by the Sylow theorems for simple groups, MathOverflow, 2021.
Index entries for sequences related to groups.

Cf. A001034, A056866, A060793, A338853.

divq[n_, p_] := AnyTrue[Rest @ Divisors[n], Mod[#, p] == 1 &]; q[1] = True; q[n_] := AllTrue[FactorInteger[n][[;; , 1]], divq[n, #] &]; Select[Range[600], q] (* Amiram Eldar, May 05 2022 *)

isok(k) = {my(f=factor(k), d=divisors(f)); for (i=1, #f~, if (vecsum(apply(x->((x % f[i,1]) == 1), d)) == 1, return(0)); ); return(1);} \\ Michel Marcus, Mar 11 2022

print([ n for n in range(1, 601)
        if set( prime_factors(n) )
        == set( p for p in prime_factors(n)
                for d in divisors(n)
                if d > 1 and d < n
                if p.divides(d - 1)
      ) ] )  # Peter Luschny, Mar 14 2022

A327912 Orders of perfect non-simple groups.

Original entry on oeis.org

120, 336, 720, 960, 1080, 1320, 1344, 1920, 2160, 2184, 2688, 3000, 3600, 3840, 4860, 4896, 5040, 5376, 5760, 6840, 7200, 7500, 7560, 7680, 9720, 10080, 10752, 11520, 12144, 14400, 14520, 14580, 15000, 15120, 15360, 15600, 16464, 17280, 19656, 20160, 21504, 21600, 23040, 24360, 28224, 29160, 29760, 30240, 30720, 32256, 34560, 37500, 39600, 40320, 43008, 43200, 43320, 43740, 46080, 48000, 50616, 51840, 56448, 57600, 57624, 58240, 58320, 60480

Offset: 1

Sébastien Palcoux, Sep 29 2019

nonn

The smallest number n such that there is a simple group and a non-simple perfect group of order n is 20160. So this sequence is A060793 minus A001034 (as sets) for the orders less than 20160. The next known such exceptions are 181440, 262080, 443520 and 604800.

The perfect groups of order 61440, 122880, 172032, 245760, 344064, 491520, 688128, 983040 have not completely been determined yet. Then GAP neither provides the number of these groups nor the groups themselves.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.
D.F. Holt and W. Plesken, Perfect Groups, Oxford Math. Monographs, Oxford University Press, 1989.

The GAP Group, GAP Data Library "Perfect Groups".

Cf. A001034, A005180, A060793.

OrderPerfectNonSimple:=function(n1,n2)
   local it,S,G,L,o,No,i,c;
   it:=SimpleGroupsIterator(n1,n2);
   S:=[];
   for G in it do
      Add(S,Order(G));
   od;
   L:=[];
   for o in [n1..n2] do
      c:=0;
      for i in S do
         if i=o then
            c:=c+1;
         fi;
      od;
      No:=NumberPerfectGroups(o);
      if No>c then
         Add(L,o);
         if c>0 then
            Print([o,c,No]);
         fi;
      fi;
   od;
   return L;
end;;

cp's OEIS Frontend

A352287 Numbers k such that, for every prime p dividing k, k has a nontrivial divisor which is congruent to 1 (mod p).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Sage