A174601 -id:A174601 - cp's OEIS frontend

A001228 Orders of sporadic simple groups.

7920, 95040, 175560, 443520, 604800, 10200960, 44352000, 50232960, 244823040, 898128000, 4030387200, 145926144000, 448345497600, 460815505920, 495766656000, 42305421312000, 64561751654400, 273030912000000, 51765179004000000, 90745943887872000, 4089470473293004800, 4157776806543360000, 86775571046077562880, 1255205709190661721292800, 4154781481226426191177580544000000, 808017424794512875886459904961710757005754368000000000

Offset: 1

N. J. A. Sloane

Numbers of divisors: A174601(n) = A000005(a(n)); squarefree kernels: A174848(n) = A007947(a(n)). - Reinhard Zumkeller, Apr 02 2010

By historical convention, the Tits group is often excluded from the list of sporadic simple groups. It could be inserted as a(7) = 17971200 giving this sequence 27 rather than 26 elements. - Charles R Greathouse IV, Jul 09 2020

			The first term is 7920 because the order of the sporadic group M_{11} is 7920, the smallest order of any sporadic group.

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].
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 296.
Martin Gardner, "The Last Recreations", 1997, chap 9, p. 153.

David Madore, Table of sporadic simple groups.
Grant Sanderson, Group theory and 808,017,424,794,512,875,886,459,904,961,710,757,005,754,368,000,000,000, 3Blue1Brown video (2020)
Eric Weisstein's World of Mathematics, Sporadic Group
Index entries for sequences related to groups

Cf. A001034, A005180.

Entries checked by Pab Ter (pabrlos(AT)yahoo.com), May 29 2004

A174670 Divisors of the order of the Monster group.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 23, 24, 25, 26, 27, 28, 29, 30, 31, 32, 33, 34, 35, 36, 38, 39, 40, 41, 42, 44, 45, 46, 47, 48, 49, 50, 51, 52, 54, 55, 56, 57, 58, 59, 60, 62, 63, 64, 65, 66, 68, 69, 70, 71, 72, 75, 76, 77, 78, 80

Offset: 1

Reinhard Zumkeller, Apr 02 2010

Let Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131;
the sequence is finite with A174601(26) = 424488960 terms;
a(n) = n for n < 37 = A053669(Mnr) = (smallest prime not in A002267);
24 of the 26 terms of A001228 are divisors of Mnr, the exceptions are A001228(19) and A001228(23), orders of groups Ly and J4;
also the first 36 factorials and the first 11 primorials are divisors of Mnr (cf. examples);
A174671 gives divisors of Mnr sorted into decreasing order: A174671(n)=a(424488960-n+1)=Mnr/a(n).

			......... a(30) = A002110(3) = ........... 30 = 5#;
........ a(101) = A000142(5) = .......... 120 = 5!;
........ a(159) = A002110(4) = .......... 210 = 7#;
........ a(398) = A000142(6) = .......... 720 = 6!;
........ a(888) = A002110(5) = ......... 2310 = 11#;
....... a(1461) = A000142(7) = ......... 5040 = 7!;
....... a(1931) = A001228(1) = ......... 7920;
....... a(4207) = A002110(6) = ........ 30030 = 13#;
....... a(4952) = A000142(8) = ........ 40320 = 8!;
....... a(7859) = A001228(2) = ........ 95040;
...... a(10787) = A001228(3) = ....... 175560;
...... a(15477) = A000142(9) = ....... 362880 = 9!;
...... a(17056) = A001228(4) = ....... 443520;
...... a(18257) = A002110(7) = ....... 510510 = 17#;
...... a(19792) = A001228(5) = ....... 604800;
...... a(44571) = A000142(10) = ..... 3628800 = 10!;
...... a(67510) = A002110(8) = ...... 9699690 = 19#;
...... a(68918) = A001228(6) = ..... 10200960;
..... a(118553) = A000142(11) = .... 39916800 = 11!;
..... a(123436) = A001228(7) = ..... 44352000;
..... a(129447) = A001228(8) = ..... 50232960;
..... a(223787) = A002110(9) = .... 223092870 = 23#;
..... a(231256) = A001228(9) = .... 244823040;
..... a(291999) = A000142(12) = ... 479001600 = 12!.
..... a(360936) = A001228(10) = ... 898128000;
..... a(584543) = A001228(11) = .. 4030387200;
.. a(424488960) = A001228(26) = ......... Mnr, the last term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Sporadic Group
Index entries for sequences related to divisors of numbers

divisors(808017424794512875886459904961710757005754368000000000)
\\ Warning: output is ~13 GB.
\\ Charles R Greathouse IV, Sep 02 2015

A051161 a(n) is the exponent of n-th prime in the order (A003131) of the Monster simple group.

Original entry on oeis.org

46, 20, 9, 6, 2, 3, 1, 1, 1, 1, 1, 0, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

nonn

a(n)=0 for n > 20.

Product_{k>0} (a(k) + 1) = A174601(26). - Reinhard Zumkeller, Apr 02 2010

Cf. A003131, A001379, A002267, A174601.

A174671 Divisors of the order of the Monster group, sorted into decreasing order.

Original entry on oeis.org

808017424794512875886459904961710757005754368000000000, 404008712397256437943229952480855378502877184000000000, 269339141598170958628819968320570252335251456000000000, 202004356198628218971614976240427689251438592000000000, 161603484958902575177291980992342151401150873600000000

Offset: 1

Reinhard Zumkeller, Apr 02 2010

Let Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131;
a(n) = Mnr / A174670(n);
the sequence is finite with A174601(26) = 424488960 terms;
a(n) = A174670(424488960 - n + 1).

			a(1) = Mnr;
a(424488960) = 1, the last term.

Reinhard Zumkeller, Table of n, a(n) for n = 1..100
Index entries for sequences related to divisors of numbers

A321224 Sporadic numbers: n is defined to be sporadic if the set of groups G not in {A_n, S_n} and having a core-free maximal subgroup of index n is nonempty and contains only sporadic simple groups.

Original entry on oeis.org

266, 506, 759, 1045, 1288, 1463, 3795

Offset: 1

Sébastien Palcoux, Aug 27 2019

A finite group G has a core-free maximal subgroup H of index n if and only if it is a primitive permutation group of degree n (acting on the set G/H of cosets).
There are no other sporadic numbers less than 4096 (see computation below).
According to Derek Holt, the next sporadic number is 4180, and the last one should be 492693551703971265784426771318116315247411200000000 (coming from the maximal subgroup 41:40 of the Monster, and assuming that L_2(13) is not maximal).
Derek Holt suggested another sequence where we also allow the extensions of the sporadic simple groups.

The GAP Group, GAP - Groups, Algorithms, and Programming, Version 4.9.3, 2018. gap-system.org.

S. Palcoux, The sporadic numbers (version: 2019-07-22), MathOverflow.

Cf. A102842, A001228, A174601, A174848, A261717, A263447, A121236.

IsSporadic:=function(G)
   if not IsSimple(G) then
      return false;
   else
      return IsomorphismTypeInfoFiniteSimpleGroup(G).series="Spor";
   fi;
end;;
SporadicNumbers:=function(b1,b2)
   local L,i,n,a,j,G;
   L:=[];
   for i in [b1..b2] do
      n:=NrPrimitiveGroups(i);
      if n>2 then
         a:=0;
         for j in [1..n] do
            G:=PrimitiveGroup(i,j);
            if not G=SymmetricGroup(i) and not G=AlternatingGroup(i) and not IsSporadic(G) then
               a:=1;
               break;
            fi;
         od;
         if a=0 then
            Add(L,i);
         fi;
      fi;
   od;
   return L;
end;;
SporadicNumbers(1,4095);
# gives: [ 266, 506, 759, 1045, 1288, 1463, 3795 ]

cp's OEIS Frontend

A001228 Orders of sporadic simple groups.

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A174670 Divisors of the order of the Monster group.

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A051161 a(n) is the exponent of n-th prime in the order (A003131) of the Monster simple group.

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A174671 Divisors of the order of the Monster group, sorted into decreasing order.

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A321224 Sporadic numbers: n is defined to be sporadic if the set of groups G not in {A_n, S_n} and having a core-free maximal subgroup of index n is nonempty and contains only sporadic simple groups.

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