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%I #5 Mar 30 2024 15:57:04
%S 1,2,3,6,7,22,23,32,33,48,49,86,87,112,113,516,517,580,581,1110,1111,
%T 1136,1137,1604,1605,5206,5207,5232,5233,5700,5701,8212,8213,9236,
%U 9237,13332,13333,16386,16387,16450,16451,17474,17475,21570,21571,24576,24577
%N Numbers whose binary indices have (1) prime indices covering an initial interval and (2) squarefree product.
%C A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798.
%C A binary index of n is any position of a 1 in its reversed binary expansion. The binary indices of n are row n of A048793.
%F Intersection of A371292 and A371289.
%e The terms together with their prime indices of binary indices begin:
%e 1: {{}}
%e 2: {{1}}
%e 3: {{},{1}}
%e 6: {{1},{2}}
%e 7: {{},{1},{2}}
%e 22: {{1},{2},{3}}
%e 23: {{},{1},{2},{3}}
%e 32: {{1,2}}
%e 33: {{},{1,2}}
%e 48: {{3},{1,2}}
%e 49: {{},{3},{1,2}}
%e 86: {{1},{2},{3},{4}}
%e 87: {{},{1},{2},{3},{4}}
%e 112: {{3},{1,2},{4}}
%e 113: {{},{3},{1,2},{4}}
%e 516: {{2},{1,3}}
%e 517: {{},{2},{1,3}}
%e 580: {{2},{4},{1,3}}
%e 581: {{},{2},{4},{1,3}}
%t normQ[m_]:=m=={}||Union[m]==Range[Max[m]];
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t Select[Range[1000],SquareFreeQ[Times @@ bpe[#]]&&normQ[Join@@prix/@bpe[#]]&]
%Y Without the covering condition we have A371289.
%Y Without squarefree product we have A371292.
%Y Interchanging binary and prime indices gives A371448.
%Y A000009 counts partitions covering initial interval, compositions A107429.
%Y A000670 counts ordered set partitions, allowing empty sets A000629.
%Y A005117 lists squarefree numbers.
%Y A011782 counts multisets covering an initial interval.
%Y A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A070939 gives length of binary expansion.
%Y A096111 gives product of binary indices.
%Y A131689 counts patterns by number of distinct parts.
%Y A302521 lists MM-numbers of set partitions, with empties A302505.
%Y A326701 lists BII-numbers of set partitions.
%Y A368533 lists numbers with squarefree binary indices, prime indices A302478.
%Y Cf. A000040, A001222, A255906, A326782, A371291, A371294, A371447, A371452.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 28 2024
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