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A multiset whose multiplicities are the prime indices of n is separable.
7

%I #9 Jul 03 2020 06:58:37

%S 1,2,4,6,8,9,12,15,16,18,20,24,25,27,30,32,35,36,40,42,45,48,49,50,54,

%T 56,60,63,64,70,72,75,77,80,81,84,90,96,98,99,100,105,108,110,112,120,

%U 121,125,126,128,132,135,140,143,144,147,150,154,160,162,165

%N A multiset whose multiplicities are the prime indices of n is separable.

%C A multiset is separable if it has a permutation that is an anti-run, meaning there are no adjacent equal parts.

%C A multiset whose multiplicities are the prime indices of n (such as row n of A305936) is not the same as the multiset of prime indices of n. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

%e The sequence together with the corresponding multisets begins:

%e 1: {}

%e 2: {1}

%e 4: {1,2}

%e 6: {1,1,2}

%e 8: {1,2,3}

%e 9: {1,1,2,2}

%e 12: {1,1,2,3}

%e 15: {1,1,1,2,2}

%e 16: {1,2,3,4}

%e 18: {1,1,2,2,3}

%e 20: {1,1,1,2,3}

%e 24: {1,1,2,3,4}

%e 25: {1,1,1,2,2,2}

%e 27: {1,1,2,2,3,3}

%e 30: {1,1,1,2,2,3}

%t nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];

%t Select[Range[100],Select[Permutations[nrmptn[#]],!MatchQ[#,{___,x_,x_,___}]&]!={}&]

%Y The complement is A335126.

%Y Anti-run compositions are A003242.

%Y Anti-runs are ranked by A333489.

%Y Separable partitions are A325534.

%Y Inseparable partitions are A325535.

%Y Separable factorizations are A335434.

%Y Inseparable factorizations are A333487.

%Y Separable partitions are ranked by A335433.

%Y Inseparable partitions are ranked by A335448.

%Y Anti-run permutations of prime indices are A335452.

%Y Patterns contiguously matched by compositions are A335457.

%Y Cf. A056239, A106351, A112798, A114938, A292884, A335489, A335516, A335838.

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 02 2020