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A213206 Largest order of permutation without a 2-cycle of n elements. Equivalently, largest LCM of partitions of n without parts =2. 1

%I #17 Feb 18 2013 07:09:16

%S 1,1,1,3,4,5,6,12,15,20,21,30,60,60,84,105,140,140,210,420,420,420,

%T 420,840,840,1260,1260,1540,1540,2520,4620,4620,5460,5460,9240,9240,

%U 13860,13860,16380,16380,27720,27720,32760,60060,60060,60060,60060,120120,120120,180180,180180,180180

%N Largest order of permutation without a 2-cycle of n elements. Equivalently, largest LCM of partitions of n without parts =2.

%H Joerg Arndt, <a href="/A213206/b213206.txt">Table of n, a(n) for n = 0..101</a>

%F a(n) = A000793(n) unless n is a term of A007504 (sum of first primes).

%e The 11 partitions (including those with parts =2) of 6 are the following:

%e [ #] [ partition ] LCM( parts )

%e [ 1] [ 1 1 1 1 1 1 ] 1

%e [ 2] [ 1 1 1 1 2 ] 2

%e [ 3] [ 1 1 1 3 ] 3

%e [ 4] [ 1 1 2 2 ] 2

%e [ 5] [ 1 1 4 ] 4

%e [ 6] [ 1 2 3 ] 6 (max, with a part =2)

%e [ 7] [ 1 5 ] 5

%e [ 8] [ 2 2 2 ] 2

%e [ 9] [ 2 4 ] 4

%e [10] [ 3 3 ] 3

%e [11] [ 6 ] 6 (max, without a part =2)

%e The largest order 6 is obtained twice, the first such partition is forbidden for this sequence, but not the second, so a(6) = A000793(6) = 6.

%e The 7 partitions (including those with parts =2) of 5 are the following:

%e [ #] [ partition ] LCM( parts )

%e [ 1] [ 1 1 1 1 1 ] 1

%e [ 2] [ 1 1 1 2 ] 2

%e [ 3] [ 1 1 3 ] 3

%e [ 4] [ 1 2 2 ] 2

%e [ 5] [ 1 4 ] 4

%e [ 6] [ 2 3 ] 6 (max with a part =2)

%e [ 7] [ 5 ] 5 (max, without a part =2)

%e The largest order (A000793(5)=6) with a part =2 is obtained with the partition into distinct primes; the largest order without a part =2 is a(5)=5.

%K nonn

%O 0,4

%A _Joerg Arndt_, Feb 15 2013

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