cp's OEIS Frontend

A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.

%I A371445 #11 Mar 31 2024 08:48:08
%S A371445 2,3,4,5,7,8,9,11,13,16,17,19,23,25,27,29,31,32,37,41,43,47,49,53,55,
%T A371445 59,61,64,65,67,71,73,79,81,83,89,97,101,103,107,109,113,115,121,125,
%U A371445 127,128,131,137,139,143,145,149,151,157,163,167,169,173,179,181
%N A371445 Numbers whose distinct prime indices are binary carry-connected and have no binary containments.
%C A371445 Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.
%C A371445 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 A371445 A binary carry of two positive integers is an overlap of binary indices. A multiset is said to be binary carry-connected iff the graph whose vertices are the elements and whose edges are binary carries is connected.
%C A371445 A binary containment is a containment of binary indices. For example, the numbers {3,5} have binary indices {{1,2},{1,3}}, so there is a binary carry but not a binary containment.
%F A371445 Intersection of A371455 and A325118.
%e A371445 The terms together with their prime indices begin:
%e A371445      2: {1}            37: {12}              97: {25}
%e A371445      3: {2}            41: {13}             101: {26}
%e A371445      4: {1,1}          43: {14}             103: {27}
%e A371445      5: {3}            47: {15}             107: {28}
%e A371445      7: {4}            49: {4,4}            109: {29}
%e A371445      8: {1,1,1}        53: {16}             113: {30}
%e A371445      9: {2,2}          55: {3,5}            115: {3,9}
%e A371445     11: {5}            59: {17}             121: {5,5}
%e A371445     13: {6}            61: {18}             125: {3,3,3}
%e A371445     16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
%e A371445     17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
%e A371445     19: {8}            67: {19}             131: {32}
%e A371445     23: {9}            71: {20}             137: {33}
%e A371445     25: {3,3}          73: {21}             139: {34}
%e A371445     27: {2,2,2}        79: {22}             143: {5,6}
%e A371445     29: {10}           81: {2,2,2,2}        145: {3,10}
%e A371445     31: {11}           83: {23}             149: {35}
%e A371445     32: {1,1,1,1,1}    89: {24}             151: {36}
%t A371445 stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
%t A371445 bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
%t A371445 prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
%t A371445 csm[s_]:=With[{c=Select[Subsets[Range[Length[s]],{2}], Length[Intersection@@s[[#]]]>0&]},If[c=={},s,csm[Sort[Append[Delete[s,List/@c[[1]]], Union@@s[[c[[1]]]]]]]]];
%t A371445 Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]
%Y A371445 Contains all powers of primes A000961 except 1.
%Y A371445 Case of A325118 (counted by A325098) without binary containments.
%Y A371445 For binary indices of binary indices we have A326750 = A326704 /\ A326749.
%Y A371445 For prime indices of prime indices we have A329559 = A305078 /\ A316476.
%Y A371445 An opposite version is A371294 = A087086 /\ A371291.
%Y A371445 Partitions of this type are counted by A371446.
%Y A371445 Carry-connected case of A371455 (counted by A325109).
%Y A371445 A001187 counts connected graphs.
%Y A371445 A007718 counts non-isomorphic connected multiset partitions.
%Y A371445 A048143 counts connected antichains of sets.
%Y A371445 A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.
%Y A371445 A070939 gives length of binary expansion.
%Y A371445 A326964 counts connected set-systems, covering A323818.
%Y A371445 Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.
%K A371445 nonn
%O A371445 1,1
%A A371445 _Gus Wiseman_, Mar 30 2024