A371452 Number 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

Offset

1,3

Comments

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.

Links

Table of n, a(n) for n=1..87.

Example

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.

Mathematica

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}]

Crossrefs

Positions of first appearances are A080355, opposite A325782.

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

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

Positions of ones are A371291.

For binary indices of prime indices we have A371451, 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.

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

A326964 counts connected set-systems, covering A323818.

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

Sequence in context: A305301 A106140 A355300 * A211097 A354907 A086342

Adjacent sequences: A371449 A371450 A371451 * A371453 A371454 A371455

Keyword

nonn

Author

Gus Wiseman, Apr 01 2024

Status

approved