cp's OEIS Frontend

A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.

%I A371447 #5 Mar 31 2024 23:51:37
%S A371447 1,2,4,5,6,8,10,12,15,16,17,18,20,24,25,26,30,32,33,34,35,36,40,42,45,
%T A371447 47,48,50,51,52,54,55,60,64,65,66,68,70,72,75,78,80,84,85,86,90,94,96,
%U A371447 99,100,102,104,105,108,110,119,120,123,125,126,127,128,130
%N A371447 Numbers whose binary indices of prime indices cover an initial interval of positive integers.
%C A371447 Also Heinz numbers of integer partitions whose parts have binary indices covering an initial interval.
%C A371447 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 A371447 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.
%e A371447 The terms together with their binary indices of prime indices begin:
%e A371447    1: {}
%e A371447    2: {{1}}
%e A371447    4: {{1},{1}}
%e A371447    5: {{1,2}}
%e A371447    6: {{1},{2}}
%e A371447    8: {{1},{1},{1}}
%e A371447   10: {{1},{1,2}}
%e A371447   12: {{1},{1},{2}}
%e A371447   15: {{2},{1,2}}
%e A371447   16: {{1},{1},{1},{1}}
%e A371447   17: {{1,2,3}}
%e A371447   18: {{1},{2},{2}}
%e A371447   20: {{1},{1},{1,2}}
%e A371447   24: {{1},{1},{1},{2}}
%e A371447   25: {{1,2},{1,2}}
%e A371447   26: {{1},{2,3}}
%e A371447   30: {{1},{2},{1,2}}
%e A371447   32: {{1},{1},{1},{1},{1}}
%t A371447 normQ[m_]:=Or[m=={},Union[m]==Range[Max[m]]];
%t A371447 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371447 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371447 Select[Range[1000],normQ[Join@@bpe/@prix[#]]&]
%Y A371447 For prime indices of prime indices we have A320456.
%Y A371447 For binary indices of binary indices we have A326754.
%Y A371447 An opposite version is A371292, A371293.
%Y A371447 The case with squarefree product of prime indices is A371448.
%Y A371447 The connected components of this multiset system are counted by A371451.
%Y A371447 A000009 counts partitions covering initial interval, compositions A107429.
%Y A371447 A000670 counts patterns, ranked by A333217.
%Y A371447 A011782 counts multisets covering an initial interval.
%Y A371447 A048793 lists binary indices, reverse A272020, length A000120, sum A029931.
%Y A371447 A070939 gives length of binary expansion.
%Y A371447 A112798 lists prime indices, reverse A296150, length A001222, sum A056239.
%Y A371447 A131689 counts patterns by number of distinct parts.
%Y A371447 Cf. A000040, A000961, A019565, A055887, A255906, A325097, A325118, A326782, A368109, A371452.
%K A371447 nonn
%O A371447 1,2
%A A371447 _Gus Wiseman_, Mar 31 2024