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A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

%I A371455 #10 May 12 2025 10:13:35
%S A371455 1,2,3,4,5,6,7,8,9,11,12,13,14,16,17,18,19,21,23,24,25,26,27,28,29,31,
%T A371455 32,33,35,36,37,38,41,42,43,47,48,49,52,53,54,55,56,57,58,59,61,63,64,
%U A371455 65,67,69,71,72,73,74,76,79,81,83,84,86,89,95,96,97,98,99
%N A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.
%C A371455 In an antichain of sets, no edge is a proper subset of any other.
%H A371455 Jakub Buczak, <a href="/A371455/b371455.txt">Table of n, a(n) for n = 1..10000</a>
%e A371455 The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
%e A371455 The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.
%t A371455 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A371455 bix[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371455 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371455 Select[Range[100],stableQ[bix/@prix[#],SubsetQ]&]
%Y A371455 Contains all powers of primes A000961.
%Y A371455 An opposite version is A087086, carry-connected case A371294.
%Y A371455 For prime indices of prime indices we have A316476, carry-connected A329559.
%Y A371455 These antichains are counted by A325109.
%Y A371455 For binary indices of binary indices we have A326704, carry-conn. A326750.
%Y A371455 The carry-connected case is A371445, counted by A371446.
%Y A371455 A048143 counts connected antichains of sets.
%Y A371455 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371455 A050320 counts set multipartitions of prime indices, see also A318360.
%Y A371455 A070939 gives length of binary expansion.
%Y A371455 A089259 counts set multipartitions of integer partitions.
%Y A371455 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371455 A116540 counts normal set multipartitions.
%Y A371455 A302478 ranks set multipartitions, cf. A073576.
%Y A371455 A325118 ranks carry-connected partitions, counted by A325098.
%Y A371455 A371451 counts carry-connected components of binary indices.
%Y A371455 Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.
%K A371455 nonn
%O A371455 1,2
%A A371455 _Gus Wiseman_, Apr 01 2024