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A096111 gives product of binary indices.

Cf. A000720, A019565, A087086, A096111, A325097, A326782, A368109, A371292, A371294, A371445, A371447.

For prime indices of prime indices we have A305079, positions of ones A305078.

For binary indices of binary indices we have A326753, positions of ones A326749.

For binary indices of prime indices we have A371451, positions of ones A325118.

A087086 lists numbers whose binary indices are pairwise indivisible.

Cf. A000040, A000720, A019565, ~A055887, ~A255906, A087086, A325097, A326782, A368109, A371292, A371294, `A371445, `A371447.

allocated for Gus WisemanNumber of connected components of the prime indices of the binary indices of n.

1, 1, 2, 1, 2, 2, 3, 1, 2, 1, 2, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 2, 3, 1, 2, 2, 3, 2, 3, 3, 4, 2, 3, 2, 3, 3, 4, 3, 4, 2, 3, 3, 4, 3, 4, 4, 5

1,3

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.

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.

The prime indices of binary indices of 281492156579880 are {{1,1},{1,2},{3,4},{4,4}}, with 2 connected components {{1,1},{1,2}} and {{3,4},{4,4}}, so a(281492156579880) = 2.

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]]]]]]]]];

bix[n_]:=Join@@Position[Reverse[IntegerDigits[n, 2]], 1];

prix[n_]:=If[n==1, {}, Flatten[Cases[FactorInteger[n], {p_, k_}:>Table[PrimePi[p], {k}]]]];

Table[Length[csm[prix/@bix[n]]], {n, 100}]

Positions of first appearances are A080355, opposite A325782.

For prime indices of prime indices we have A305079, positions of ones A305078.

For binary indices of binary indices we have A326753, positions of ones A326749.

Positions of ones are A371291.

For binary indices of prime indices we have A371451, positions of ones A325118.

A001187 counts connected graphs.

A007718 counts non-isomorphic connected multiset partitions.

A048143 counts connected antichains of sets.

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

A070939 gives length of binary expansion.

A087086 lists numbers whose binary indices are pairwise indivisible.

A096111 gives product of binary indices.

A112798 lists prime indices, reverse A296150, length A001222, sum A056239.

A326964 counts connected set-systems, covering A323818.

Cf. A000040, A000720, A019565, ~A055887, ~A255906, A325097, A326782, A368109, A371292, A371294, `A371445, `A371447.

allocated

nonn

Gus Wiseman, Apr 01 2024

approved

editing

allocated for Gus Wiseman

allocated

approved