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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-François Alcover, Mar 23 2017, translated from Maple *) CROSSREFS Column k=1 of A258119. Cf. A002033, A215366, A259941. Sequence in context: A333021 A114874 A100361 * A069654 A330359 A000068 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 August 1 00:13 EDT 2021. Contains 346377 sequences. (Running on oeis4.)