%I #11 Mar 31 2024 08:48:08

%S 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 59,61,64,65,67,71,73,79,81,83,89,97,101,103,107,109,113,115,121,125,

%U 127,128,131,137,139,143,145,149,151,157,163,167,169,173,179,181

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

%C Also Heinz numbers of binary carry-connected integer partitions whose distinct parts have no binary containments, counted by A371446.

%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 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 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 Intersection of A371455 and A325118.

%e The terms together with their prime indices begin:

%e 2: {1} 37: {12} 97: {25}

%e 3: {2} 41: {13} 101: {26}

%e 4: {1,1} 43: {14} 103: {27}

%e 5: {3} 47: {15} 107: {28}

%e 7: {4} 49: {4,4} 109: {29}

%e 8: {1,1,1} 53: {16} 113: {30}

%e 9: {2,2} 55: {3,5} 115: {3,9}

%e 11: {5} 59: {17} 121: {5,5}

%e 13: {6} 61: {18} 125: {3,3,3}

%e 16: {1,1,1,1} 64: {1,1,1,1,1,1} 127: {31}

%e 17: {7} 65: {3,6} 128: {1,1,1,1,1,1,1}

%e 19: {8} 67: {19} 131: {32}

%e 23: {9} 71: {20} 137: {33}

%e 25: {3,3} 73: {21} 139: {34}

%e 27: {2,2,2} 79: {22} 143: {5,6}

%e 29: {10} 81: {2,2,2,2} 145: {3,10}

%e 31: {11} 83: {23} 149: {35}

%e 32: {1,1,1,1,1} 89: {24} 151: {36}

%t stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];

%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 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 Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

%Y Contains all powers of primes A000961 except 1.

%Y Case of A325118 (counted by A325098) without binary containments.

%Y For binary indices of binary indices we have A326750 = A326704 /\ A326749.

%Y For prime indices of prime indices we have A329559 = A305078 /\ A316476.

%Y An opposite version is A371294 = A087086 /\ A371291.

%Y Partitions of this type are counted by A371446.

%Y Carry-connected case of A371455 (counted by A325109).

%Y A001187 counts connected graphs.

%Y A007718 counts non-isomorphic connected multiset partitions.

%Y A048143 counts connected antichains of sets.

%Y A048793 lists binary indices, A000120 length, A272020 reverse, A029931 sum.

%Y A070939 gives length of binary expansion.

%Y A326964 counts connected set-systems, covering A323818.

%Y Cf. A019565, A056239, A112798, A304713, A304716, A305079, A305148, A325097, A325105, A325107, A325119, A371452.

%K nonn

%O 1,1

%A _Gus Wiseman_, Mar 30 2024