A371455 Numbers k such that if we take the binary indices of each prime index of k we get an antichain of sets.

1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 16, 17, 18, 19, 21, 23, 24, 25, 26, 27, 28, 29, 31, 32, 33, 35, 36, 37, 38, 41, 42, 43, 47, 48, 49, 52, 53, 54, 55, 56, 57, 58, 59, 61, 63, 64, 65, 67, 69, 71, 72, 73, 74, 76, 79, 81, 83, 84, 86, 89, 95, 96, 97, 98, 99

Offset

1,2

Comments

In an antichain of sets, no edge is a proper subset of any other.

Links

Table of n, a(n) for n=1..68.

Example

The prime indices of 65 are {3,6} with binary indices {{1,2},{2,3}} so 65 is in the sequence.
The prime indices of 255 are {2,3,7} with binary indices {{2},{1,2},{1,2,3}} so 255 is not in the sequence.

Mathematica

stableQ[u_, Q_]:=!Apply[Or, Outer[#1=!=#2&&Q[#1, #2]&, u, u, 1], {0, 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}]]]];
Select[Range[100], stableQ[bix/@prix[#], SubsetQ]&]

Crossrefs

Contains all powers of primes A000961.

An opposite version is A087086, carry-connected case A371294.

For prime indices of prime indices we have A316476, carry-connected A329559.

These antichains are counted by A325109.

For binary indices of binary indices we have A326704, carry-conn. A326750.

The carry-connected case is A371445, counted by A371446.

A048143 counts connected antichains of sets.

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

A050320 counts set multipartitions of prime indices, see also A318360.

A070939 gives length of binary expansion.

A089259 counts set multipartitions of integer partitions.

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

A116540 counts normal set multipartitions.

A302478 ranks set multipartitions, cf. A073576.

A325118 ranks carry-connected partitions, counted by A325098.

A371451 counts carry-connected components of binary indices.

Cf. A019565, A050326, A304713, A305148, A325097, A325107, A371291, A371452.

Sequence in context: A122154 A122156 A336547 * A325097 A333126 A330697

Adjacent sequences: A371452 A371453 A371454 * A371456 A371457 A371458

Keyword

nonn

Author

Gus Wiseman, Apr 01 2024

Status

approved