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

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, 59, 61, 64, 65, 67, 71, 73, 79, 81, 83, 89, 97, 101, 103, 107, 109, 113, 115, 121, 125, 127, 128, 131, 137, 139, 143, 145, 149, 151, 157, 163, 167, 169, 173, 179, 181

Offset: 1

Author

Gus Wiseman, Mar 30 2024

Keywords

• nonn

Comments

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

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.

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.

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.

Examples

			The terms together with their prime indices begin:
     2: {1}            37: {12}              97: {25}
     3: {2}            41: {13}             101: {26}
     4: {1,1}          43: {14}             103: {27}
     5: {3}            47: {15}             107: {28}
     7: {4}            49: {4,4}            109: {29}
     8: {1,1,1}        53: {16}             113: {30}
     9: {2,2}          55: {3,5}            115: {3,9}
    11: {5}            59: {17}             121: {5,5}
    13: {6}            61: {18}             125: {3,3,3}
    16: {1,1,1,1}      64: {1,1,1,1,1,1}    127: {31}
    17: {7}            65: {3,6}            128: {1,1,1,1,1,1,1}
    19: {8}            67: {19}             131: {32}
    23: {9}            71: {20}             137: {33}
    25: {3,3}          73: {21}             139: {34}
    27: {2,2,2}        79: {22}             143: {5,6}
    29: {10}           81: {2,2,2,2}        145: {3,10}
    31: {11}           83: {23}             149: {35}
    32: {1,1,1,1,1}    89: {24}             151: {36}
		

Crossrefs

Contains all powers of primes A000961 except 1.

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

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

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

An opposite version is A371294 = A087086 /\ A371291.

Partitions of this type are counted by A371446.

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

A001187 counts connected graphs.

A007718 counts non-isomorphic connected multiset partitions.

A048143 counts connected antichains of sets.

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

A070939 gives length of binary expansion.

A326964 counts connected set-systems, covering A323818.

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

Programs

Mathematica

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
bpe[n_]:=Join@@Position[Reverse[IntegerDigits[n,2]],1];
prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n], {p_,k_}:>Table[PrimePi[p],{k}]]]];
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]]]]]]]]];
Select[Range[100],stableQ[bpe/@prix[#],SubsetQ] && Length[csm[bpe/@prix[#]]]==1&]

Formula

Intersection of A371455 and A325118.