A301900 -id:A301900 - cp's OEIS frontend

A305613 Numbers whose multiset of prime factors is not knapsack.

30, 60, 70, 72, 84, 90, 120, 140, 144, 150, 168, 180, 210, 216, 240, 252, 270, 280, 286, 288, 300, 308, 330, 336, 350, 360, 378, 390, 420, 432, 440, 450, 480, 490, 495, 504, 510, 525, 528, 540, 560, 570, 572, 576, 588, 594, 600, 616, 630, 646, 648, 660, 672

Offset: 1

Gus Wiseman, Jun 06 2018

nonn

A multiset of positive integers is knapsack if every distinct submultiset has a different sum.

			30 = 2 * 3 * 5 is not knapsack because 2 + 3 = 5.

Cf. A027746, A108917, A122768, A275972, A276024, A299701, A299729, A301855, A301900, A304793, A305611.

Select[Range[1000],DivisorSigma[0,#]=!=Length[Union[Total/@Subsets[Join@@Cases[FactorInteger[#],{p_,k_}:>Table[p,{k}]]]]]&]

A316361 FDH numbers of strict integer partitions such that not every distinct subset has a different average.

Original entry on oeis.org

24, 56, 60, 110, 120, 135, 140, 168, 210, 216, 224, 264, 270, 273, 280, 308, 312, 315, 330, 342, 360, 378, 384, 408, 420, 440, 456, 459, 480, 504, 520, 540, 546, 550, 552, 576, 585, 594, 600, 616, 630, 660, 672, 693, 696, 702, 728, 744, 756, 759, 760, 770, 780

Offset: 1

Gus Wiseman, Jun 30 2018

nonn

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).

			210 is the FDH number of (5,4,2,1), and the subsets {1,5}, and {2,4} have the same average, so 210 belongs to the data.

Cf. A050376, A064547, A108917, A213925, A275972, A299755, A299757, A301899, A301900, A316271, A316313, A316362.

nn=1000;
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],!UnsameQ@@Mean/@Union[Subsets[FDfactor[#]/.FDrules]]&]

A316401 Number of strict integer partitions of n that are not knapsack (not every subset has a different sum) but every subset has a different average.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 1, 0, 2, 0, 1, 0, 3, 1, 3, 2, 4, 2, 6, 6, 11, 9, 9, 10, 20, 16, 18, 17, 27, 24, 31, 29, 43, 31, 43, 40, 59, 52, 58, 61, 83, 68, 93, 80, 124, 99, 120, 109, 145, 151, 185, 160, 232, 163, 257, 229, 314, 280, 286, 310, 427, 385, 513, 333, 596

Offset: 1

Gus Wiseman, Jul 01 2018

nonn

			The a(22) = 11 partitions:
(11,6,5), (11,7,4), (11,8,3), (11,9,2), (11,10,1),
(11,5,4,2), (11,6,3,2), (11,8,2,1), (12,5,3,2), (12,5,4,1), (14,4,3,1).

Cf. A000009, A108917, A275972, A299702, A301899, A301900, A316271, A316313, A316314, A316399, A316400.

Table[Length[Select[IntegerPartitions[n],UnsameQ@@#&&!UnsameQ@@Total/@Union[Subsets[#]]&&UnsameQ@@Mean/@Union[Subsets[#]]&]],{n,20}]

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A305613 Numbers whose multiset of prime factors is not knapsack.

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A316361 FDH numbers of strict integer partitions such that not every distinct subset has a different average.

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A316401 Number of strict integer partitions of n that are not knapsack (not every subset has a different sum) but every subset has a different average.

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Author

Keywords

Examples

Crossrefs

Programs

Mathematica