A051704 -id:A051704 - cp's OEIS frontend

A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

1, 1, 1, 3, 3, 5, 5, 7, 15, 15, 21, 21, 35, 35, 60, 105, 105, 105, 140, 210, 210, 420, 420, 420, 420, 840, 1155, 1260, 1365, 1540, 2310, 2520, 4620, 4620, 5460, 5460, 9240, 9240, 13860, 15015, 16380, 16380, 27720, 30030, 32760, 60060, 60060, 60060

Offset: 0

Vladeta Jovovic

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].
V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Index entries for sequences related to groups

Cf. A057742, A057743, A057740, A000793.

(* a3 = A000793  a4 = A051704 *) a3[n_] := Max[LCM @@@ IntegerPartitions[n]]; a4[n_] := (pp = Reap[ Do[ pk = p^k; If[pk <= n, Sow[pk]], {p, Prime[ Range[2, PrimePi[n]]]}, {k, 1, Ceiling[ Log[3, n]]}]][[2, 1]]; sel = Select[ IntegerPartitions[n, All, pp], Length[#] == Length[ Union[#] && !MatchQ[#, {_, x_, _, y_, _} /; GCD[x, y] != 1]] &]; Max[Times @@@ sel]); a4[0] = 1; a4[1] = a4[2] = a4[4] = a4[6] = 0; a[n_] := Max[a3[n - 2], a4[n - 1], a4[n]]; a[0] = a[1] = a[2] = 1; Table[a[n], {n, 0, 47}] (* Jean-François Alcover, Sep 11 2012, from formula *)

a(n)={my(m=1); forpart(p=n, if(sum(i=1, #p, p[i]-1)%2==0, m=max(m, lcm(Vec(p))))); m} \\ Andrew Howroyd, Jul 03 2018

a(n)=max{ A000793(n-2), A051704(n-1), A051704(n) }, a(0)=a(1)=1.

A051703 Maximal value of products of partitions of n into powers of distinct primes (1 not considered a power).

Original entry on oeis.org

1, 0, 2, 3, 4, 6, 0, 12, 15, 20, 30, 28, 60, 40, 84, 105, 140, 210, 180, 420, 280, 330, 360, 840, 504, 1260, 1155, 1540, 2310, 2520, 4620, 3080, 5460, 3960, 9240, 5544, 13860, 6552, 16380, 15015, 27720, 30030, 32760, 60060, 40040, 45045, 51480, 120120

Offset: 0

Vladeta Jovovic

nonn

			a(11) = 28 because max{11, 2*3^2, 2^3*3, 2^2*7} = 28.

Robert Gerbicz, Table of n, a(n) for n = 0..1000
J. Bamberg, G. Cairns and D. Kilminster, The crystallographic restriction, permutations and Goldbach's conjecture, Amer. Math. Monthly, 110 (March 2003), 202-209.

Largest element of n-th row of A080743.
A000793(n)=max{A000793(n-1), a(n)}, A000793(0)=1.
Cf. A008475, A051613, A080743, A080744, A051704.

b:= proc(n, i) option remember; local p;
      p:= `if`(i<1, 1, ithprime(i));
      `if`(n=0, 1, `if`(i<1 or n<0, 0, max(b(n, i-1),
      seq(p^j*b(n-p^j, i-1), j=1..ilog[p](n))) ))
    end:
a:= n-> b(n, numtheory[pi](n)):
seq(a(n), n=0..60);  # Alois P. Heinz, Feb 16 2013

nmax = 48; Do[a[n]=0, {n, 1, nmax}]; km = PrimePi[nmax]; For[k=1, k <= km, k++, q = 1; p = Prime[k]; For[i=nmax, i >= 1, i--, q=1; While[q*p <= i, q *= p; If[i == q, m = q, If[a[i - q] != 0, m = q*a[i - q], m = 0]]; a[i] = Max[a[i], m]]]]; a[0] = 1; Table[a[n], {n, 0, nmax}] (* Jean-François Alcover, Aug 02 2012, translated from Robert Gerbicz's Pari program *)

{N=1000;v=vector(N,i,0);forprime(p=2,N,q=1;forstep(i=N,1,-1,
q=1;while(q*p<=i,q*=p;if(i==q,M=q,if(v[i-q],M=q*v[i-q],M=0));
v[i]=max(v[i],M))));print(0" "1);for(i=1,N,print(i" "v[i]))} \\ Robert Gerbicz, Jul 31 2012

Corrected and extended by Robert Gerbicz, Jul 31 2012

cp's OEIS Frontend

A051593 Largest order of even permutation of n elements, or maximal order of element of alternating group A_n.

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