A113907 -id:A113907 - cp's OEIS frontend

A162020 Positive k such that there is at least one partition into parts {14, 52, 78, 133, 248} (terms of A113907).

14, 28, 42, 52, 56, 66, 70, 78, 80, 84, 92, 94, 98, 104, 106, 108, 112, 118, 120, 122, 126, 130, 132, 133, 134, 136, 140, 144, 146, 147, 148, 150, 154, 156, 158, 160, 161, 162, 164, 168, 170, 172, 174, 175, 176, 178, 182, 184, 185, 186, 188, 189, 190, 192, 196, 198, 199, 200, 202, 203, 204, 206, 208, 210, 211, 212

Offset: 1

Jonathan Vos Post, Jun 24 2009

All integers greater than 327 are also in the sequence. There are 163 values less than 327, and 164 in the complement of this sequence. The first value arising in two different ways is 156 = 52 + 52 + 52 = 78 + 78. The first triples of three consecutive values: (132,133,134), (160,161,162). "Use the Frobenius problem. Since gcd(52,133)=1 the maximum value not in the sequence is 52*133-52-133=6731 even using only 52 and 133, by a simple dp code you can check all values up to this limit and in fact n=327 is the maximum not in the original sequence." -- Robert Gerbicz.

Wikipedia, Coin problem.

Cf. A113907.

L := [14,52,78,133,248] ; # check whether n is a linear combination of op(..,L) with # some nonnegative a,b,c,d,e frob := proc(n,L) local a,b,c,d,e,nres ; for a from 0 do if a*op(1,L) > n then break; fi; for b from 0 do if a*op(1,L)+b*op(2,L) > n then break; fi; for c from 0 do if a*op(1,L)+b*op(2,L)+c*op(3,L) > n then break; fi; for d from 0 do if a*op(1,L)+b*op(2,L)+c*op(3,L)+d*op(4,L) > n then break; fi; nres := n-a*op(1,L)-b*op(2,L)-c*op(3,L)-d*op(4,L) ; if nres < 0 then break; fi; if nres = 0 then RETURN([a,b,c,d,0]) ; fi; if ( nres mod op(5,L) ) = 0 then RETURN([a,b,c,d,nres/op(5,L)]) ; fi; od: od; od: od: RETURN([]) ; end: for n from 14 to 200 do f := frob(n,L) ; if f <> [] then print(n,f) ; fi; od: Extended and b-file by R. J. Mathar.

ok[n_] := Reduce[a >= 0 && b >= 0 && c >= 0 && d >= 0 && e >= 0 && 14a + 52b + 78c + 133d + 248e == n, {a, b, c, d, e}, Integers] =!= False; Select[Range[196], ok] (* Jean-François Alcover, Sep 06 2011 *)

Better name from Joerg Arndt, May 23 2021

A003134 Orders of Weyl groups of type E_n.

Original entry on oeis.org

51840, 2903040, 696729600

Offset: 6

N. J. A. Sloane

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. 120.
H. S. M. Coxeter and W. O. J. Moser, Generators and Relations for Discrete Groups, 4th ed., Springer-Verlag, NY, reprinted 1984.

Wikipedia, Exceptional Lie algebra

Cf. A001217, A113907.

[ #CoxeterGroup(Sprintf("E%o",n)) : n in [6..8]]; // Sergei Haller (sergei(AT)sergei-haller.de), Dec 21 2006

A334597 Dimensions of the finite-dimensional Lie algebras of type E_n (n=3,...,8).

Original entry on oeis.org

11, 24, 45, 78, 133, 248

Offset: 3

Bart Vlaar, May 07 2020

E_3 = A_1 A_2, E_4 = A_4, E_5 = D_5.

R. Carter, Lie Algebras of Finite and Affine Type, Cambridge University Press, 2005, 561-609.

Wikipedia, E_n (Lie algebra)

The last three terms are the last three terms of A113907.

For n>3, a(n) = 2*A272764(n) + n (for n=3 the Lie algebra is not simple).

cp's OEIS Frontend

A162020 Positive k such that there is at least one partition into parts {14, 52, 78, 133, 248} (terms of A113907).

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A003134 Orders of Weyl groups of type E_n.

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A334597 Dimensions of the finite-dimensional Lie algebras of type E_n (n=3,...,8).

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