A051161 -id:A051161 - cp's OEIS frontend

A001379 Degrees of irreducible representations of Monster group M.

1, 196883, 21296876, 842609326, 18538750076, 19360062527, 293553734298, 3879214937598, 36173193327999, 125510727015275, 190292345709543, 222879856734249, 1044868466775133, 1109944460516150, 2374124840062976, 8980616927734375, 8980616927734375, 15178147608537368

Offset: 1

N. J. A. Sloane

The sequence contains 194 terms, of which 170 are distinct. The only triple of repeated terms is a(123) = a(124) = a(125) = 5514132424881463208443904. The rest of the repeated terms are pairs, for example a(16) = a(17) = 8980616927734375. - Omar E. Pol, Nov 28 2014

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].

Eric M. Schmidt, Table of n, a(n) for n = 1..194 (complete sequence)
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
John F. R. Duncan, Michael J. Griffin and Ken Ono, Proof of the Umbral Moonshine Conjecture, arXiv:1503.01472 [math.RT], 2015.
Yang-Hui He, John McKay, Sporadic and Exceptional, arXiv:1505.06742 [math.AG], 2015.
Valdo Tatitscheff, A short introduction to Monstrous Moonshine, arXiv:1902.03118 [math.NT], 2019.
Eric Weisstein's World of Mathematics, Monstrous Moonshine
Index entries for sequences related to groups

Cf. A003131, A002267, A051161.

List(Irr(CharacterTable("M")), chi->chi[1]); # Eric M. Schmidt, Jul 15 2012

A002267 The 15 supersingular primes: primes dividing order of Monster simple group.

Original entry on oeis.org

2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71

Offset: 1

N. J. A. Sloane

The supersingular primes are a subset of the Chen primes (A109611). - Paul Muljadi, Oct 12 2005

PROD(a(k): 1<=k<=15) = 1618964990108856390 = A174848(26). - Reinhard Zumkeller, Apr 02 2010

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].
A. P. Ogg, Modular functions, in The Santa Cruz Conference on Finite Groups (Univ. California, Santa Cruz, Calif., 1979), pp. 521-532, Proc. Sympos. Pure Math., 37, Amer. Math. Soc., Providence, R.I., 1980.

J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067 [math.QA], 2001.
Alan W. Reid, Arithmetic hyperbolic manifolds, slides of a talk, Cornell University, June 2014,
G. K. Sankaran, A supersingular coincidence, arXiv:2009.11379 [math.NT], 2020.
J. G. Thompson, Finite groups and modular functions, Bulletin of the London Mathematical Society 11.3 (1979): 347-351. See page 350.
Eric Weisstein's World of Mathematics, Supersingular Prime
Index entries for sequences related to groups

Cf. A003131, A001379, A051161, A109611.

FactorInteger[GroupOrder[MonsterGroupM[]]][[All, 1]] (* Jean-François Alcover, Oct 03 2016 *)

A002267=vecextract(primes(20),612351) \\ bitmask 2^20-1-213<<11: remove primes # 12, 14, 16, 18 and 19. - M. F. Hasler, Nov 10 2017

A003131 Order of Monster simple group.

Original entry on oeis.org

8, 0, 8, 0, 1, 7, 4, 2, 4, 7, 9, 4, 5, 1, 2, 8, 7, 5, 8, 8, 6, 4, 5, 9, 9, 0, 4, 9, 6, 1, 7, 1, 0, 7, 5, 7, 0, 0, 5, 7, 5, 4, 3, 6, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 54

N. J. A. Sloane

			808017424794512875886459904961710757005754368000000000.

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], p. 228.
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 296.
J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, New York, 1996, p. 62.
C. A. Pickover, The Math Book, Sterling, NY, 2009; see p. 474.

Cf. A001228, A001379, A002267, A051161, A174670, A174817.

2^46 * 3^20 * 5^9 * 7^6 * 11^2 * 13^3 * 17 * 19 * 23 * 29 * 31 * 41 * 47 * 59 * 71 \\ Charles R Greathouse IV, Oct 31 2014

Equals A001228(26) = Product_{n=1..15} A002267(n)^A051161(A049084(A002267(n))) = Sum_{n=0..53} a(53-n)*10^n = h(53) with h(n) = 10*h(n-1) + a(n) for n > 0, h(0) = a(0). - Reinhard Zumkeller, Apr 02 2010

A174601 Numbers of divisors of orders of sporadic simple groups.

Original entry on oeis.org

60, 112, 128, 192, 192, 384, 480, 384, 704, 896, 1056, 1920, 2688, 3200, 2816, 4256, 4320, 5880, 16128, 16896, 25536, 26400, 45056, 143616, 1580544, 424488960

Offset: 1

Reinhard Zumkeller, Apr 02 2010

a(n) = A000005(A001228(n)).

			a(1) = A000005(7920) = A000005(2^4 * 3^2 * 5^1 * 11^1) = (4+1)*(2+1)*(1+1)*(1+1) = 60;
a(26) = PROD((A051161(k)+1): k>0) = (46+1)*(20+1)*(9+1)*(6+1)*(2+1)*(3+1)*(1+1)*(1+1)*(1+1)*(1+1)*(1+1)*(0+1)*(1+1)*(0+1)*(1+1)*(0+1)*(1+1)*(0+1)*(0+1)*(1+1)*(0+1)*(0+1)*... = 47*21*10*7*3*4*2*2*2*2*2*1*2*1*2*1*2*1*1*2*1*1*... = 424488960.

Eric Weisstein's World of Mathematics, Sporadic Group

Cf. A174670.

A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

Original entry on oeis.org

1, 1, 1, 2, 12, 96, 1152, 16128, 290304, 6967296, 181149696, 5796790272, 208684449792, 7930009092096, 333060381868032, 15986898329665536, 863292509801938944, 48344380548908580864, 2997351594032332013568

Offset: 1

Labos Elemer, Feb 28 2001

nonn

Such products arise in Hardy-Littlewood prime k-tuplet conjectural formulas.

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 84-94.
R. K. Guy, Unsolved Problems in Number Theory, A8, A1
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979.
G. Polya, Mathematics and Plausible Reasoning, Vol. II, Appendix Princeton UP, 1954

G. C. Greubel, Table of n, a(n) for n = 1..350
C. K. Caldwell, Prime k-tuple Conjecture
Steven R. Finch, Hardy-Littlewood Constants [Broken link]
Steven R. Finch, Hardy-Littlewood Constants [From the Wayback machine]
G. H. Hardy and J. E. Littlewood, Some problems of 'partitio numerorum'; III: on the expression of a number as a sum of primes, Acta Mathematica, Vol. 44, pp. 1-70, 1923.
G. Niklasch, Some number theoretical constants: 1000-digit values [Cached copy]
G. Polya, Heuristic reasoning in the theory of numbers, Am. Math. Monthly,66 (1959), 375-384.

Cf. A000847, A002110, A005867, A022008, A048298, A049296, A051160, A051161, A051162.
Cf. A051163, A051164, A051165, A051166, A051167, A051168, A059861, A059862, A059863.
Cf. A059864, A059865.

[n le 3 select 1 else (&*[NthPrime(j) -5: j in [4..n]]): n in [1..30]]; // G. C. Greubel, Feb 02 2023

Join[{1,1,1},FoldList[Times,Prime[Range[4,20]]-5]] (* Harvey P. Dale, Dec 29 2018 *)

a(n) = prod(k=4, n, prime(k)-5); \\ Michel Marcus, Dec 12 2017

def A059864(n): return product(nth_prime(j) -5 for j in range(4,n+1))
[A059864(n) for n in range(1,31)] # G. C. Greubel, Feb 02 2023

A343743 a(n) is the largest base in which the order of the Monster group has (47 - n) zeros; alternatively, radicals of maximal powers dividing the order of the Monster group.

Original entry on oeis.org

2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 4, 4, 12, 12, 12, 12, 12, 24, 24, 24, 24, 48, 144, 1440, 1440, 2880, 120960, 1451520, 87091200, 1902071808000, 15184923989114880000, 808017424794512875886459904961710757005754368000000000

Offset: 1

Hal M. Switkay, Jun 27 2021

Let z be a specified minimum number of zeros in the order of the Monster group; here z is a natural number, 1 <= z <= 46, with z = (47 - n). Then the largest base in which the order of the Monster group has at least z zeros is:
Product_{k=1..20} prime(k)^floor(A051161(k)/z).
When z = 1 this is the order of the Monster group.
Every term in this sequence except the last is a number of least prime signature (A025487).
In the following table, when the order of the Monster group has exactly z zeros, it also has s significant digits, and d = s + z total digits.
   z   s   d
  -- --- ---
  46 134 180
  23  67  90
  20  30  50
  15  25  40
  11  22  33
  10  15  25
   9   9  18
   7   9  16
   6   5  11
   5   4   9
   4   3   7
   3   2   5
   2   1   3
   1   1   2

			a(27) = the largest base in which the order of the Monster group has at least (47 - 27) = 20 zeros. This is 2^(floor(46/20)) * 3^(floor(20/20)) = 2^2 * 3 = 12; the remaining terms in the product have exponent 0.

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, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Cf. A051161.

f = FactorInteger[MonsterGroupM[] // GroupOrder]; Table[Times @@ ((First[#]^Floor[Last[#]/z]) & /@ f), {z, Max[f[[;; , 2]]], 1, -1}] (* Amiram Eldar, Jul 19 2021 *)

A348092 Unique values, or record values, of A343743.

Original entry on oeis.org

2, 4, 12, 24, 48, 144, 1440, 2880, 120960, 1451520, 87091200, 1902071808000, 15184923989114880000, 808017424794512875886459904961710757005754368000000000

Offset: 1

Hal M. Switkay, Sep 29 2021

Every term in this sequence except the last is a number of least prime signature (A025487).
In the following table, when the order of the Monster group is written in base a(n), it has exactly z zeros, s significant digits, and d = s + z total digits.
   n  z   s   d
  -- -- --- ---
46 134 180
23  67  90
20  30  50
15  25  40
11  22  33
10  15  25
9   9  18
7   9  16
6   5  11
5   4   9
4   3   7
3   2   5
2   1   3
1   1   2
a(n) is the largest natural number b such that the order of the Monster group is divisible by b^z.

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, N. J. A. Sloane, Sphere Packings, Lattices, and Groups. Springer, 3rd ed., 1999.

Cf. A051161, A343743.

f = FactorInteger[MonsterGroupM[] // GroupOrder]; DeleteDuplicates@ Table[Times @@ ((First[#]^Floor[Last[#]/z]) & /@ f), {z, Max[f[[;; , 2]]], 1, -1}] (* Amiram Eldar, Sep 30 2021 *)

a(n) = Product_{k=1..20} prime(k)^floor(A051161(k)/z(n)).

cp's OEIS Frontend

A001379 Degrees of irreducible representations of Monster group M.

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A002267 The 15 supersingular primes: primes dividing order of Monster simple group.

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A003131 Order of Monster simple group.

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A174601 Numbers of divisors of orders of sporadic simple groups.

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A059864 a(n) = Product_{i=4..n} (prime(i)-5), where prime(i) is i-th prime.

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A343743 a(n) is the largest base in which the order of the Monster group has (47 - n) zeros; alternatively, radicals of maximal powers dividing the order of the Monster group.

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Mathematica

A348092 Unique values, or record values, of A343743.

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