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%I #9 Jul 03 2020 06:58:23
%S 3,5,7,10,11,13,14,17,19,21,22,23,26,28,29,31,33,34,37,38,39,41,43,44,
%T 46,47,51,52,53,55,57,58,59,61,62,65,66,67,68,69,71,73,74,76,78,79,82,
%U 83,85,86,87,88,89,91,92,93,94,95,97,101,102,103,104,106
%N A multiset whose multiplicities are the prime indices of n is inseparable.
%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 of terms together with the corresponding multisets begins:
%e 3: {1,1}
%e 5: {1,1,1}
%e 7: {1,1,1,1}
%e 10: {1,1,1,2}
%e 11: {1,1,1,1,1}
%e 13: {1,1,1,1,1,1}
%e 14: {1,1,1,1,2}
%e 17: {1,1,1,1,1,1,1}
%e 19: {1,1,1,1,1,1,1,1}
%e 21: {1,1,1,1,2,2}
%e 22: {1,1,1,1,1,2}
%e 23: {1,1,1,1,1,1,1,1,1}
%e 26: {1,1,1,1,1,1,2}
%e 28: {1,1,1,1,2,3}
%e 29: {1,1,1,1,1,1,1,1,1,1}
%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 A335127.
%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. A025487, A056239, A106351, A112798, A114938, A181819, A181821, A278990, A292884, A335407, A335489, A335516, A335838.
%K nonn
%O 1,1
%A _Gus Wiseman_, Jul 01 2020
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