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

1,1

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).

Links

Table of n, a(n) for n=1..53.

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]&]

Crossrefs

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

Sequence in context: A026040 A259217 A211567 * A357699 A328375 A141717

Adjacent sequences: A316268 A316269 A316270 * A316272 A316273 A316274

Keyword

nonn

Author

Gus Wiseman, Jun 28 2018

Status

approved