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A316271
FDH numbers of strict non-knapsack partitions.
6
24, 40, 70, 84, 120, 126, 135, 168, 198, 210, 216, 220, 231, 264, 270, 280, 286, 312, 330, 351, 360, 364, 378, 384, 408, 416, 420, 440, 456, 462, 504, 520, 528, 540, 544, 546, 552, 560, 576, 594, 600, 616, 630, 640, 646, 660, 663, 680, 696, 702, 728, 744, 748
OFFSET
1,1
COMMENTS
A strict integer partition is knapsack if every subset has a different sum.
Let f(n) = A050376(n) be the n-th Fermi-Dirac prime. The FDH number of a strict integer partition (y_1,...,y_k) is f(y_1)*...*f(y_k).
EXAMPLE
a(1) = 24 is the FDH number of (3,2,1), which is not knapsack because 3 = 2 + 1.
MATHEMATICA
nn=1000;
sksQ[ptn_]:=And[UnsameQ@@ptn, UnsameQ@@Plus@@@Union[Subsets[ptn]]];
FDfactor[n_]:=If[n==1, {}, Sort[Join@@Cases[FactorInteger[n], {p_, k_}:>Power[p, Cases[Position[IntegerDigits[k, 2]//Reverse, 1], {m_}->2^(m-1)]]]]];
FDprimeList=Array[FDfactor, nn, 1, Union]; FDrules=MapIndexed[(#1->#2[[1]])&, FDprimeList];
Select[Range[nn], !sksQ[FDfactor[#]/.FDrules]&]
KEYWORD
nonn
AUTHOR
Gus Wiseman, Jun 28 2018
STATUS
approved