A225655 a(n) = largest LCM of partitions of n divisible by n.

1, 2, 3, 4, 5, 6, 7, 8, 9, 30, 11, 60, 13, 84, 105, 16, 17, 180, 19, 420, 420, 330, 23, 840, 25, 780, 27, 1540, 29, 4620, 31, 32, 4620, 3570, 9240, 13860, 37, 7980, 16380, 27720, 41, 32760, 43, 60060, 45045, 19320, 47, 55440, 49, 23100, 157080, 180180, 53

Offset: 1

Author

Antti Karttunen, May 19 2013

Keywords

• nonn

Comments

a(n) = lcm(p1,p2,...,pk) for that partition of n for which the LCM is a multiple of n, and which maximizes this value among all such partitions [p1,p2,...,pk] of n.

Links

• Alois P. Heinz, Table of n, a(n) for n = 1..100 (terms n = 0..83 from Antti Karttunen)
• Index entries for sequences related to lcm's

Crossrefs

For all n, a(A225651(n)) = A000793(A225651(n)).

Cf. A225646, A225656, A225652.

A225657 lists the values of n for which a(n) = n.

Programs

Maple

b:= proc(n, i) option remember; `if`(n=0, {1},
      `if`(i<1, {}, {seq(map(x->ilcm(x, `if`(j=0, 1, i)),
       b(n-i*j, i-1))[], j=0..n/i)}))
    end:
a:= n-> max(select(x-> irem(x, n)=0, b(n$2))[]):
seq(a(n), n=1..50);  # Alois P. Heinz, May 26 2013

Mathematica

b[n_, i_] := b[n, i] = If[n==0, {1}, If[i<1, {}, Union @ Flatten @ Table[ Map[ Function[{x}, LCM[x, If[j==0, 1, i]]], b[n-i*j, i-1]], {j, 0, n/i}]]]; a[n_] := Max[Select[b[n, n], Mod[#, n]==0&]]; Table[a[n], {n, 1, 50}] (* Jean-François Alcover, Jul 29 2015, after Alois P. Heinz *)