A174670 Divisors of the order of the Monster group.

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

1,2

Comments

Let Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131.

The sequence is finite with A174601(26) = 424488960 terms;

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).

The first positive integers missing in this sequence are the primes 37, 43, 53, 61, 67 and 73. - M. F. Hasler, Apr 30 2026

Links

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

Eric Weisstein's World of Mathematics, Monster Group.

Index entries for sequences related to divisors of numbers

Formula

a(n) = n for n < 37 = A053669(Mnr) = smallest prime not in A002267.

Example

......... 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.

Mathematica

PositionIndex[Divisible[GroupOrder[MonsterGroupM[]], Range[100]]][True] (* Paolo Xausa, May 04 2026 *)

Prog

(PARI) divisors(808017424794512875886459904961710757005754368000000000)
\\ Warning: output is ~13 GB.
\\ Charles R Greathouse IV, Sep 02 2015
(Python)
def divisors(factored={2:46, 3:20, 5:9, 7:6, 11:2, 13:3, 17:1,
             19:1, 23:1, 29:1, 31:1, 41:1, 47:1, 59:1, 71:1}):
    fact = sorted(factored.items()); heap = [(1, len(fact)-1, 0)]
    while heap:
        d, i, e = heapq.heappop(heap); yield d; p, m = fact[i]
        if e < m: heapq.heappush(heap, (d*p, i, e+1))
        for j in range(i):
            heapq.heappush(heap, (d*fact[j][0], j, 1))
import heapq; from itertools import islice
def A174670_first(N=424488960): return(list(islice(divisors(), N))) # M. F. Hasler, Apr 30 2026, with contribution from Martin Fuller

Crossrefs

Sequence in context: A258068 A273885 A363287 * A212554 A178772 A367141

Adjacent sequences: A174667 A174668 A174669 * A174671 A174672 A174673

Keyword

fini,nonn,easy

Author

Reinhard Zumkeller, Apr 02 2010

Status

approved