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Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n.
3

%I #15 Mar 23 2017 04:15:10

%S 1,2,4,6,16,18,64,42,100,162,1024,234,4096,1088,1936,798,65536,2300,

%T 262144,4698,18496,31744,4194304,8658,234256,167936,52900,46784,

%U 268435456,90992,1073741824,42294,984064,3866624,5345344,140300,68719476736,17563648,6885376

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

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

%H Alois P. Heinz, <a href="/A259939/b259939.txt">Table of n, a(n) for n = 0..3000</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPartition.html">Perfect Partition</a>

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

%e 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.

%p b:= (n, l)-> `if`(n=1, 2^(l[1]-1)*mul(ithprime(mul(l[j],

%p j=1..i-1))^(l[i]-1), i=2..nops(l)), min(seq(b(n/d,

%p [l[], d]), d=numtheory[divisors](n) minus{1}))):

%p a:= n-> `if`(n=0, 1, b(n+1, [])):

%p seq(a(n), n=0..42);

%t 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}}]]];

%t a[n_] := If[n==0, 1, b[n+1, {}]];

%t Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)

%Y Column k=1 of A258119.

%Y Cf. A002033, A215366, A259941.

%K nonn

%O 0,2

%A _Alois P. Heinz_, Jul 09 2015