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

Author

Gus Wiseman, Jul 04 2022

Keywords

• nonn

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, ...

Examples

			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}
		

Links

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

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.

Programs

Mathematica

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

Extensions

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