A259939 Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n.

1, 2, 4, 6, 16, 18, 64, 42, 100, 162, 1024, 234, 4096, 1088, 1936, 798, 65536, 2300, 262144, 4698, 18496, 31744, 4194304, 8658, 234256, 167936, 52900, 46784, 268435456, 90992, 1073741824, 42294, 984064, 3866624, 5345344, 140300, 68719476736, 17563648, 6885376

Offset: 0

Author

Alois P. Heinz, Jul 09 2015

Keywords

• nonn

Comments

A perfect partition of n contains a unique partition for any k in {0,...,n}. See also A002033.

Examples

			For n=7 there are 4 perfect partitions: [4,1,1,1], [4,2,1], [2,2,2,1] and [1,1,1,1,1,1,1], their encodings as Product_{i:lambda} prime(i) give 56, 42, 54, 128, respectively.  The smallest value is a(7) = 42.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..3000
• Eric Weisstein's World of Mathematics, Perfect Partition

Crossrefs

Column k=1 of A258119.

Cf. A002033, A215366, A259941.

Programs

Maple

b:= (n, l)-> `if`(n=1, 2^(l[1]-1)*mul(ithprime(mul(l[j],
      j=1..i-1))^(l[i]-1), i=2..nops(l)), min(seq(b(n/d,
        [l[], d]), d=numtheory[divisors](n) minus{1}))):
a:= n-> `if`(n=0, 1, b(n+1, [])):
seq(a(n), n=0..42);

Mathematica

b[n_, l_] := If[n==1, 2^(l[[1]]-1)*Product[Prime[Product[l[[j]], {j, 1, i-1}]]^(l[[i]]-1), {i, 2, Length[l]}], Min[Table[b[n/d, Append[l, d]], {d, Divisors[n] ~Complement~ {1}}]]];
a[n_] := If[n==0, 1, b[n+1, {}]];
Table[a[n], {n, 0, 42}] (* Jean-François Alcover, Mar 23 2017, translated from Maple *)

Formula

a(n) = A258119(n,1).