OFFSET
1,1
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.
EXAMPLE
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}
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&]
CROSSREFS
Contains all powers of primes A000961 except 1.
Partitions of this type are counted by A371446.
A001187 counts connected graphs.
A007718 counts non-isomorphic connected multiset partitions.
A048143 counts connected antichains of sets.
A070939 gives length of binary expansion.
KEYWORD
nonn
AUTHOR
Gus Wiseman, Mar 30 2024
STATUS
approved