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%I #4 Mar 31 2024 23:51:32
%S 1,2,4,5,6,8,10,12,15,16,17,20,24,26,30,32,33,34,40,47,48,51,52,55,60,
%T 64,66,68,80,85,86,94,96,102,104,110,120,123,127,128,132,136,141,143,
%U 160,165,170,172,187,188,192,204,205,208,215,220,221,226,240,246
%N Numbers such that (1) the product of prime indices is squarefree, and (2) the binary indices of prime indices cover an initial interval of positive integers.
%C Also Heinz numbers of integer partitions whose parts have (1) squarefree product and (2) binary indices covering an initial interval.
%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.
%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.
%F Intersection of A302505 and A371447.
%e The terms together with their binary indices of prime indices begin:
%e 1: {}
%e 2: {{1}}
%e 4: {{1},{1}}
%e 5: {{1,2}}
%e 6: {{1},{2}}
%e 8: {{1},{1},{1}}
%e 10: {{1},{1,2}}
%e 12: {{1},{1},{2}}
%e 15: {{2},{1,2}}
%e 16: {{1},{1},{1},{1}}
%e 17: {{1,2,3}}
%e 20: {{1},{1},{1,2}}
%e 24: {{1},{1},{1},{2}}
%e 26: {{1},{2,3}}
%e 30: {{1},{2},{1,2}}
%e 32: {{1},{1},{1},{1},{1}}
%e 33: {{2},{1,3}}
%e 34: {{1},{1,2,3}}
%e 40: {{1},{1},{1},{1,2}}
%e 47: {{1,2,3,4}}
%e 48: {{1},{1},{1},{1},{2}}
%e 51: {{2},{1,2,3}}
%t normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t Select[Range[1000], SquareFreeQ[Times@@prix[#]]&&normQ[Join@@bpe/@prix[#]]&]
%Y An opposite version is A371293, A371292.
%Y Without the squarefree condition we have A371447, see also A320456, A326754.
%Y The connected components of this multiset system are counted by A371451.
%Y A000009 counts partitions covering initial interval, compositions A107429.
%Y A000670 counts patterns, ranked by A333217.
%Y A011782 counts multisets covering an initial interval.
%Y A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A070939 gives length of binary expansion.
%Y A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A131689 counts patterns by number of distinct parts.
%Y Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.
%K nonn
%O 1,2
%A _Gus Wiseman_, Mar 31 2024
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