A002267 -id:A002267 - cp's OEIS frontend

A108764 Products of exactly two supersingular primes (A002267).

4, 6, 9, 10, 14, 15, 21, 22, 25, 26, 33, 34, 35, 38, 39, 46, 49, 51, 55, 57, 58, 62, 65, 69, 77, 82, 85, 87, 91, 93, 94, 95, 115, 118, 119, 121, 123, 133, 141, 142, 143, 145, 155, 161, 169, 177, 187, 203, 205, 209, 213, 217, 221, 235, 247, 253, 287, 289, 295, 299

Offset: 1

Jonathan Vos Post, Jun 17 2005

There are exactly 15 supersingular primes: 2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59 and 71 (A002267). The supersingular primes are exactly the set of primes that divide the group order of the Monster group.

Peter Luschny's link shows how this sequence may be connected to Schinzel-Sierpinski conjecture and the Calkin-Wilf tree.

			1207 = 17 * 71, 3337 = 47 * 71.

E. Dijkstra, Selected Writings on Computing, Springer, 1982, p. 232.
Ogg, A. P. "Modular Functions." In The Santa Cruz Conference on Finite Groups. Held at the University of California, Santa Cruz, Calif., 1979 (Ed. B. Cooperstein and G. Mason). Providence, RI: Amer. Math. Soc., pp. 521-532, 1980.
Silverman, J. H. The Arithmetic of Elliptic Curves II. New York: Springer-Verlag, 1994.

T. D. Noe, Table of n, a(n) for n = 1..120
N. Calkin and H. S. Wilf, Recounting the rationals, Amer. Math. Monthly, 107 (No. 4, 2000), pp. 360-363.
Matthew M. Conroy, A sequence related to a conjecture of Schinzel, J. Integ. Seqs. Vol. 4 (2001), #01.1.7.
J. H. Conway and S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
J. H. Conway, R. K. Guy, W. A. Schneeberger and N. J. A. Sloane, The Primary Pretenders, Acta Arith. 78 (1997), 307-313.
P. D. T. A. Elliott, The multiplicative group of rationals generated by the shifted primes. I., J. Reine Angew. Math. 463 (1995), 169-216.
P. D. T. A. Elliott, The multiplicative group of rationals generated by the shifted primes. II. J. Reine Angew. Math. 519 (2000), 59-71.
Peter Luschny, The Schinzel-Sierpinski conjecture and the Calkin-Wilf tree.
A. Schinzel and W. Sierpinski, Sur certaines hypotheses concernant les nombres premiers, Acta Arithmetica 4 (1958), 185-208; erratum 5 (1958) p. 259.
Eric Weisstein et al., Supersingular Prime.

Cf. A001358, A002267.

Union[ Times @@@ Tuples[{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}, 2]] (*Robert G. Wilson v, Feb 11 2011 *)

{a(n)} = {p*q: p in A002267 and q in A002267}.

A212554 Products of supersingular primes (A002267).

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, 81, 82, 84, 85, 87, 88, 90, 91, 92, 93, 94, 95, 96, 98, 99, 100

Offset: 1

Ben Branman, May 21 2012

nonn

The initial 1 is included because it has no non-supersingular prime factors.

Many of the early terms divide the order of the monster simple group (see A174670). The first n such that a(n) does not belong to A174670 is a(204)=289.

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 S. P. Norton, Monstrous Moonshine, Bull. Lond. Math. Soc. 11 (1979) 308-339.
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.

T. D. Noe, Table of n, a(n) for n = 1..10000
T. Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, arXiv:math/0109067.
Eric Weisstein's World of Mathematics, Supersingular Prime

Cf. A002267, A174670, A108764 (products of exactly two supersingular primes).

ps = {2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}; fQ[n_] := Module[{p = Transpose[FactorInteger[n]][[1]]}, Complement[p, ps] == {}]; Join[{1}, Select[Range[2,1000], fQ]] (* T. D. Noe, May 21 2012 *)

log a(n) ~ k*n^(1/15). - Charles R Greathouse IV, Jul 18 2012

A212582 Products of exactly three supersingular primes (A002267).

Original entry on oeis.org

8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99, 102, 105, 110, 114, 116, 117, 124, 125, 130, 138, 147, 153, 154, 164, 165, 170, 171, 174, 175, 182, 186, 188, 190, 195, 207, 230, 231, 236, 238, 242, 245, 246, 255, 261

Offset: 1

Jonathan Vos Post, May 21 2012

The smallest "triprime" or "3-almost prime" not in this sequence is 148 = 2 * 2 * 37, as 37 is smallest prime which is not a supersingular prime.

T. D. Noe, Table of n, a(n) for n = 1..680 (complete sequence)

Cf. A002267, A014612, A108764 (products of exactly two supersingular primes), A212554 (products of supersingular primes).

Union[Times @@@ Tuples[{2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 41, 47, 59, 71}, 3]] (* T. D. Noe, May 21 2012 *)

i * j * k such that i, j, k are each in A002267, not necessarily distinct.

A001379 Degrees of irreducible representations of Monster group M.

Original entry on oeis.org

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

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

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

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.

A174848 Squarefree kernels of orders of sporadic simple groups.

Original entry on oeis.org

330, 330, 43890, 2310, 210, 53130, 2310, 9690, 53130, 2310, 3570, 79170, 30030, 1360590, 53130, 53130, 30030, 43890, 177521190, 1607970, 11741730, 690690, 75992317170, 340510170, 325046311590, 1618964990108856390

Offset: 1

Reinhard Zumkeller, Apr 02 2010

a(n) = A007947(A001228(n)).

			a(1) = 2*3*5*11;
a(2) = 2*3*5*11;
a(3) = 2*3*5*7*11*19;
a(4) = 2*3*5*7*11 = 11#;
a(5) = 2*3*5*7 = 7#;
a(6) = 2*3*5*7*11*23;
a(7) = 2*3*5*7*11 = 11#;
a(8) = 2*3*5*17*19;
a(9) = 2*3*5*7*11*23;
a(10) = 2*3*5*7*11 = 11#;
a(11) = 2*3*5*7*17;
a(12) = 2*3*5*7*13*29;
a(13) = 2*3*5*7*11*13 = 13#;
a(14) = 2*3*5*7*11*19*31;
a(15) = 2*3*5*7*11*23;
a(16) = 2*3*5*7*11*23;
a(17) = 2*3*5*7*11*13 = 13#;
a(18) = 2*3*5*7*11*19;
a(19) = 2*3*5*7*11*31*37*67;
a(20) = 2*3*5*7*13*19*31;
a(21) = 2*3*5*7*11*13*17*23;
a(22) = 2*3*5*7*11*13*23;
a(23) = 2*3*5*7*11*23*29*31*37*43;
a(24) = 2*3*5*7*11*13*17*23*29;
a(25) = 2*3*5*7*11*13*17*19*23*31*47;
a(26) = PROD(A002267(k): 1<=k<=15) = 2*3*5*7*11*13*17*19*23*29*31*41*47*59*71.

Eric Weisstein's World of Mathematics, Sporadic Group

Cf. A174670.

A173105 The 15 supersingular primes written in octal.

Original entry on oeis.org

2, 3, 5, 7, 13, 15, 21, 23, 27, 35, 37, 51, 57, 73, 107

Offset: 1

Jonathan Vos Post, Feb 09 2010

The supersingular primes are the primes that divide the order of the Monster group.

Cf. A000040, A002267, A007094.

a(n) = A007094(A002267(n)).

cp's OEIS Frontend

A108764 Products of exactly two supersingular primes (A002267).

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A212554 Products of supersingular primes (A002267).

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A212582 Products of exactly three supersingular primes (A002267).

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A001379 Degrees of irreducible representations of Monster group M.

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A174670 Divisors of the order of the Monster group.

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A003131 Order of Monster simple 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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A174848 Squarefree kernels of orders of sporadic simple groups.

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A173105 The 15 supersingular primes written in octal.

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