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

0,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..3000

Eric Weisstein's World of Mathematics, Perfect Partition

Formula

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

Example

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.

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 *)

Crossrefs

Column k=1 of A258119.

Cf. A002033, A215366, A259941.

Sequence in context: A392006 A114874 A100361 * A069654 A330359 A000068

Adjacent sequences: A259936 A259937 A259938 * A259940 A259941 A259942

Keyword

nonn

Author

Alois P. Heinz, Jul 09 2015

Status

approved