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A259939 Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n. 3
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 (list; graph; refs; listen; history; text; internal format)
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-Fran├žois Alcover, Mar 23 2017, translated from Maple *)

CROSSREFS

Column k=1 of A258119.

Cf. A002033, A215366, A259941.

Sequence in context: A287681 A114874 A100361 * A069654 A000068 A067662

Adjacent sequences:  A259936 A259937 A259938 * A259940 A259941 A259942

KEYWORD

nonn

AUTHOR

Alois P. Heinz, Jul 09 2015

STATUS

approved

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Last modified November 21 22:35 EST 2019. Contains 329383 sequences. (Running on oeis4.)