A316271 FDH numbers of strict non-knapsack partitions.

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

Author

Gus Wiseman, Jun 28 2018

Keywords

• nonn

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).

Examples

			a(1) = 24 is the FDH number of (3,2,1), which is not knapsack because 3 = 2 + 1.
		

Crossrefs

Cf. A000712, A005117, A050376, A056239, A064547, A108917, A213925, A275972, A284640, A299702, A299755, A299757, A301899, A301900.

Programs

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]&]