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

%I A259939 #16 Feb 16 2025 08:33:26
%S A259939 1,2,4,6,16,18,64,42,100,162,1024,234,4096,1088,1936,798,65536,2300,
%T A259939 262144,4698,18496,31744,4194304,8658,234256,167936,52900,46784,
%U A259939 268435456,90992,1073741824,42294,984064,3866624,5345344,140300,68719476736,17563648,6885376
%N A259939 Smallest Product_{i:lambda} prime(i) for any perfect partition lambda of n.
%C A259939 A perfect partition of n contains a unique partition for any k in {0,...,n}.  See also A002033.
%H A259939 Alois P. Heinz, <a href="/A259939/b259939.txt">Table of n, a(n) for n = 0..3000</a>
%H A259939 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPartition.html">Perfect Partition</a>
%F A259939 a(n) = A258119(n,1).
%e A259939 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 A259939 b:= (n, l)-> `if`(n=1, 2^(l[1]-1)*mul(ithprime(mul(l[j],
%p A259939       j=1..i-1))^(l[i]-1), i=2..nops(l)), min(seq(b(n/d,
%p A259939         [l[], d]), d=numtheory[divisors](n) minus{1}))):
%p A259939 a:= n-> `if`(n=0, 1, b(n+1, [])):
%p A259939 seq(a(n), n=0..42);
%t A259939 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 A259939 a[n_] := If[n==0, 1, b[n+1, {}]];
%t A259939 Table[a[n], {n, 0, 42}] (* _Jean-François Alcover_, Mar 23 2017, translated from Maple *)
%Y A259939 Column k=1 of A258119.
%Y A259939 Cf. A002033, A215366, A259941.
%K A259939 nonn
%O A259939 0,2
%A A259939 _Alois P. Heinz_, Jul 09 2015