A003131 -id:A003131 - cp's OEIS frontend

A174817 Near primes to Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131.

808017424794512875886459904961710757005754367999999957, 808017424794512875886459904961710757005754367999999947, 808017424794512875886459904961710757005754368000000083, 808017424794512875886459904961710757005754367999999803, 808017424794512875886459904961710757005754368000000283

Offset: 1

Reinhard Zumkeller, Apr 02 2010

nonn

Sorted by increasing distance to Mnr = abs(A174818(n)).

			a(1) = Mnr - 43 = 808017424794512875886459904961710757005754367999999957 is the nearest prime to Mnr;
a(3) = Mnr + 83 = 808017424794512875886459904961710757005754368000000083 is the smallest prime greater than Mnr; remarkably, (a(143),a(141)) = (Mnr-9511,Mnr-9509) is a twin prime pair.

Reinhard Zumkeller, Table of n, a(n) for n = 1..146
Dario A. Alpern, Factorization using the Elliptic Curve Method
Eric Weisstein's World of Mathematics, Sporadic Group

Cf. A001228, A003131, A174818.

With[{mnr=808017424794512875886459904961710757005754368000000000},SortBy[ {#,Abs[ #-mnr]}&/@Table[NextPrime[mnr,n],{n,{-4,-3,-2,-1,1,2,3,4}}],Last]][[All,1]] (* Harvey P. Dale, Nov 14 2021 *)

a(n) = Mnr + A174818(n).

a(5) aligned with b-file by Georg Fischer, Jul 11 2022

A174818 a(n) = A174817(n) - Mnr; where Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131.

Original entry on oeis.org

-43, -53, 83, -197, 283, -313, 431, -433, -439, -673, -733, 823, 881, 997, 1061, -1093, -1123, 1223, 1303, 1307, 1327, 1381, -1451, 1453, -1471, -1531, 1549, 1583, -1607, -1667, 1709, 1721, -1787, 1787, 1949, -1973, 1993, 2039, 2083, -2099, 2129

Offset: 1

Reinhard Zumkeller, Apr 02 2010

sign

The absolute values of the terms are non-divisors of Mnr (complement of A174670); the smallest composite term is ABS(a(43))=2479=37*67.

Reinhard Zumkeller, Table of n, a(n) for n = 1..146

Cf. A003131, A001228, A174670, A174817.

A191555 a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).

Original entry on oeis.org

1, 2, 12, 720, 3628800, 144850083840000, 272760108249915378892800000000, 1264767303092594444142256488682840323816161280000000000000000

Offset: 0

Rick L. Shepherd, Jun 06 2011

x^(2^n) - a(n) is the minimal polynomial over Q for the algebraic number sqrt(p(1)*sqrt(p(2)*...*sqrt(p(n-1)*sqrt(p(n)))...)), where p(k) is the k-th prime.  Each such monic polynomial is irreducible by Eisenstein's Criterion (using p = p(n)).
A prime version of Somos's quadratic recurrence sequence A052129(n) = A052129(n-1)^2 * n = Product_{k=1..n} k^(2^(n-k)). - Jonathan Sondow, Mar 29 2014
All positive integers have unique factorizations into powers of distinct primes, and into powers of squarefree numbers with distinct exponents that are powers of 2. (See A329332 for a description of the relationship between the two.) a(n) is the least number such that both factorizations have n factors. - Peter Munn, Dec 15 2019
From Peter Munn, Jan 24 2020 to Feb 06 2020: (Start)
For n >= 0, a(n+1) is the n-th power of 12 in the monoid defined by A306697.
a(n) is the least positive integer that cannot be expressed as the product of fewer than n terms of A072774 (powers of squarefree numbers).
All terms that are less than the order of the Monster simple group (A003131) are divisors of the group's order, with a(6) exceeding its square root.
(End)
It is remarkable that 4 of the first 5 terms are factorials. - Hal M. Switkay, Jan 21 2025

			a(1) = 2^1 = 2 and x^2 - 2 is the minimal polynomial for the algebraic number sqrt(2).
a(4) = 2^8*3^4*5^2*7^1 = 3628800 and x^16 - 3628800 is the minimal polynomial for the algebraic number sqrt(2*sqrt(3*sqrt(5*sqrt(7)))).

Alois P. Heinz, Table of n, a(n) for n = 0..11

Sequences with related definitions: A006939, A052129, A191554, A239350 (and thence A239349), A252738, A266639.
Cf. also A000040, A000079, A003131, A072774, A329332, A331592.
A000290, A003961, A059896, A306697 are used to express relationship between terms of this sequence.
Subsequence of A025487, A138302, A225547, A267117 (apart from a(1) = 2), A268375, A331593.
Antidiagonal products of A329050.

[n le 1 select 2 else Self(n-1)^2*NthPrime(n): n in [1..10]]; // Vincenzo Librandi, Feb 06 2016

a:= proc(n) option remember;
      `if`(n=0, 1, a(n-1)^2*ithprime(n))
    end:
seq(a(n), n=0..8);  # Alois P. Heinz, Mar 05 2020

RecurrenceTable[{a[1] == 2, a[n] == a[n-1]^2 Prime[n]}, a, {n, 10}] (* Vincenzo Librandi, Feb 06 2016 *)
Table[Product[Prime[k]^2^(n-k),{k,n}],{n,0,10}] (* or *) nxt[{n_,a_}]:={n+1,a^2 Prime[n+1]}; NestList[nxt,{0,1},10][[All,2]] (* Harvey P. Dale, Jan 07 2022 *)

a(n) = prod(k=1, n, prime(k)^(2^(n-k)))

;; Two variants, both with memoization-macro definec.
(definec (A191555 n) (if (= 1 n) 2 (* (A000040 n) (A000290 (A191555 (- n 1)))))) ;; After the original recurrence.
(definec (A191555 n) (if (= 1 n) 2 (* (A000079 (A000079 (- n 1))) (A003961 (A191555 (- n 1)))))) ;; After the alternative recurrence - Antti Karttunen, Feb 06 2016

For n > 0, a(n) = a(n-1)^2 * prime(n); a(0) = 1. [edited to extend to a(0) by Peter Munn, Feb 13 2020]
a(0) = 1; for n > 0, a(n) = 2^(2^(n-1)) * A003961(a(n-1)). - Antti Karttunen, Feb 06 2016, edited Feb 13 2020 because of the new prepended starting term.
For n > 1, a(n) = A306697(a(n-1),12) = A059896(a(n-1)^2, A003961(a(n-1))). - Peter Munn, Jan 24 2020

a(0) added by Peter Munn, Feb 13 2020

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

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

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

A096151 Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem.

Original entry on oeis.org

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

Offset: 206545

Lekraj Beedassy, Jul 27 2004

The number has 206545 digits. Archimedes's cattle problem, in equation form, requires the smallest sum W+X+Y+Z+w+x+y+z of the system W = (1/2 + 1/3)*X + Z; X = (1/4 + 1/5)*Y + Z; Y = (1/6 + 1/7)*W + Z; w = (1/3 + 1/4)*(X+x); x = (1/4 + 1/5)*(Y+y); y = (1/5 + 1/6)*(Z+z); z = (1/6 + 1/7)*(W+w), subject to the conditions that W+X be a square and Y+Z be triangular.
This in turn reduces to computing the value 224571490814418*t(1)^2, where (s(1), t(1)) is the smallest nontrivial solution to s^2 - D*t^2 = 1, with D=410286423278424 (or smallest solution t divisible by 9314 for squarefree D=4729494). [First number changed resulting from answers to a code golf challenge regarding this sequence by Jonathan Oswald, Jun 25 2020]
The final 100 digits are 0303265435652072678728835 1384925616695438960481550 0599463014429250035488311 8973723406626719455081800. - Robert G. Wilson v, Sep 02 2004. [See link below.]

A. Amthor, "Das Problema bovinum des Archimedes", Zeitschrift f. Math. u. Physik (Hist.-litt.Abtheilung), Vol. XXV (1880), pp 153-171.
D. Barthe, "Le problème des boeufs du Soleil", Les équations algébriques, pp. 134-9 Tangente Hors série No. 22 Pole Paris 2005.
A. H. Beiler, Recreations in the Theory of Numbers, pp. 249-251, Dover NY 1966.
E. T. Bell, The Last Problem, pp. 148-152, MAA Washington DC 1990.
K. Devlin, All The Math That's Fit To Print, pp. 64, MAA Washington DC 1994.
L. E. Dickson, History of the Theory of Numbers, Vol.II, pp. 342-5, Chelsea NY 1992.
H. Doerrie, 100 Great Problems of Elementary Mathematics, Prob.1, "Archimedes' Problema Bovinum", pp. 3-7 Dover NY 1965.
A. P. Domoryad, Mathematical Games and Pastimes, pp. 29-30 Pergamon Press NY 1963.
P. Haber, Mathematical Puzzles and Pastimes, Prob. 113, pp. 40-1; 60-3, The Peter Pauper Press NY 1957.
P. Hoffman, Archimedes' Revenge, pp. 29-32 Penguin 1988.
M. Klamkin, ed., Problems in Applied Mathematics: Selections from SIAM Review, SIAM, 1990; see pp. v-vi.
H. L. Nelson, "A Solution to Archimedes' Cattle Problem", Journal of Recreational Mathematics, Vol. 13:3 (1980-81), pp. 162-176.
D. Olivastro, Ancient Puzzles, "Archimedes Revenge", pp. 184-7, Bantam Books NY 1993.
M. Petkovic, "Archimedes Cattle Problem", Famous Puzzles of Great Mathematicians, pp. 41-3, Amer. Math. Soc.(AMS), Providence RI 2009.
W. L. Schaaf, Recreational Mathematics: A Guide To Literature, p. 31, NCTM Washington DC 1963.
A. Weil, Number Theory, An approach through history from Hammurapi to Legendre, pp. 18-19, Birkhäuser Boston 2001.
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, pp. 187 (Entry 4729494) Penguin Books 1987.

Robert G. Wilson v, Table of n, a(n) for n = 206545..413089
Robert G. Wilson v, Complete decimal expansion of the number (complete sequence, but not in b-file format).
Anonymous, The Archimedian Cattle Problem
Alex Bellos and Brady Haran, The Archimedes Number Numberphile video (2019)
E. Brown, Three Connections to Continued Fractions:Archimedes and the Cattle (pages 6-7/12)
B. Carroll, Archimedes and Large Numbers: Cattle Puzzle
Code Golf and Coding Challenges Challenge, Archimedes's cattle problem
K. Devlin, The Archimedes Cattle Problem
H. W. Lenstra Jr., Solving the Pell Equation, Notices of the AMS, Vol.49, No.2, Feb. 2002, p.182-192.
I. Peterson, Mathtrek, Cattle of the Sun
T. Rike, Archimedes Cattle Problem
C. Rorres, The Cattle Problem
Ian Stewart, Counting the Cattle of the Sun, Mathematical Recreations Column, Scientific American pp. 112, Apr 03 2000.
Ilan Vardi, Archimedes' Cattle Problem, Amer. Math. Month. Vol. 105(4) April 1998 pp. 305-319, MAA Washington DC.
A. Veling, Solution To Archimedes' Cattle Problem (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
A. Veling, Full Solution Printout (Copy on web.archive.org as of Oct. 2007; page does not exist anymore).
Eric Weisstein's World of Mathematics, Archimedes' Cattle Problem
H. C. Williams, R. A. German, and C. R. Zarnke, Solution of the cattle problem of Archimedes, Mathematics of Computation, Vol. XIX (1965), pp. 671-687.
A. Winans, Archimedes' Cattle Problem and Pell's Equation

See A003131 for another example of a sequence with a large offset based on a large integer. - N. J. A. Sloane, Dec 25 2018

PellSolve[(m_Integer)?Positive] := Module[{cf, n, s}, cf = ContinuedFraction[ Sqrt[m]]; n = Length[ Last[cf]]; If[ OddQ[n], n = 2*n]; s = FromContinuedFraction[ ContinuedFraction[ Sqrt[m], n]]; {Numerator[s], Denominator[s]}]; x = 4729494; y = PellSolve[x]; z = Floor[25194541/184119152(y[[1]] + y[[2]]*Sqrt[x])^4658]; Take[ IntegerDigits[z], 105] (* Robert G. Wilson v, Sep 02 2004, using A. Winans's formula *)

More terms from Robert G. Wilson v, Jul 30 2004
Reference added and two links fixed by William Rex Marshall, Nov 17 2010
Edited (broken links fixed, historical references added) by M. F. Hasler, Feb 13 2013
Offset corrected by N. J. A. Sloane, Dec 25 2018

cp's OEIS Frontend

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

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A174817 Near primes to Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131.

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A174818 a(n) = A174817(n) - Mnr; where Mnr = A001228(26) = 808017424794512875886459904961710757005754368000000000, also called the Monster number, cf. A003131.

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A191555 a(n) = Product_{k=1..n} prime(k)^(2^(n-k)).

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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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A174670 Divisors of the order of the Monster group.

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A096151 Decimal expansion of the 206545-digit integer solution to Archimedes's cattle problem.

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