A355392 Sorted positions of first appearances in A181591 = binomial(bigomega(n), omega(n)).

1, 4, 8, 16, 24, 32, 48, 96, 128, 192, 240, 256, 384, 480, 512, 768, 960, 1536, 1920, 2048, 3072, 3360, 3840, 4096, 6144, 6720, 7680, 8192, 12288, 13440, 15360, 16384, 24576, 26880, 30720, 49152, 53760, 61440, 65536, 73920, 107520, 122880, 131072, 147840, 196608

Offset

1,2

Comments

These are the positions of terms in A181591 that are different from all prior terms.

The statistic omega = A001221 counts distinct prime factors (without multiplicity).

The statistic bigomega = A001222 counts prime factors with multiplicity.

We have A181591(2^k) = k, so the image under A181591 is a permutation of the positive integers. It begins: 1, 2, 3, 4, 6, 5, 10, 15, 7, 21, 20, ...

Links

Amiram Eldar, Table of n, a(n) for n = 1..210

Example

The terms together with their prime indices begin:
    1: {}
    4: {1,1}
    8: {1,1,1}
   16: {1,1,1,1}
   24: {1,1,1,2}
   32: {1,1,1,1,1}
   48: {1,1,1,1,2}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  192: {1,1,1,1,1,1,2}
  240: {1,1,1,1,2,3}
  256: {1,1,1,1,1,1,1,1}
  384: {1,1,1,1,1,1,1,2}
  480: {1,1,1,1,1,2,3}
  512: {1,1,1,1,1,1,1,1,1}
  768: {1,1,1,1,1,1,1,1,2}
  960: {1,1,1,1,1,1,2,3}

Mathematica

s=Table[Binomial[PrimeOmega[n], PrimeNu[n]], {n, 1000}];
Select[Range[Length[s]], FreeQ[Take[s, #-1], s[[#]]]&]

Crossrefs

The unsorted version with multiplicity is A355386.

This is the sorted version of A355391.

A000005 counts divisors.

A001221 counts prime indices without multiplicity.

A001222 count prime indices with multiplicity.

A070175 gives representatives for bigomega and omega, triangle A303555.

Cf. A000712, A022811, A056239, A071625, A118914, A181819, A323023, A355384.

Sequence in context: A181823 A308985 A046059 * A290498 A137932 A309141

Adjacent sequences: A355389 A355390 A355391 * A355393 A355394 A355395

Keyword

nonn

Author

Gus Wiseman, Jul 04 2022

Extensions

a(41)-a(45) from Amiram Eldar, Jul 10 2022

Status

approved