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Sorted positions of first appearances in A181591 = binomial(bigomega(n), omega(n)).
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%I #13 Jul 10 2022 13:22:11

%S 1,4,8,16,24,32,48,96,128,192,240,256,384,480,512,768,960,1536,1920,

%T 2048,3072,3360,3840,4096,6144,6720,7680,8192,12288,13440,15360,16384,

%U 24576,26880,30720,49152,53760,61440,65536,73920,107520,122880,131072,147840,196608

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

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

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

%C The statistic bigomega = A001222 counts prime factors with multiplicity.

%C 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, ...

%H Amiram Eldar, <a href="/A355392/b355392.txt">Table of n, a(n) for n = 1..210</a>

%e The terms together with their prime indices begin:

%e 1: {}

%e 4: {1,1}

%e 8: {1,1,1}

%e 16: {1,1,1,1}

%e 24: {1,1,1,2}

%e 32: {1,1,1,1,1}

%e 48: {1,1,1,1,2}

%e 96: {1,1,1,1,1,2}

%e 128: {1,1,1,1,1,1,1}

%e 192: {1,1,1,1,1,1,2}

%e 240: {1,1,1,1,2,3}

%e 256: {1,1,1,1,1,1,1,1}

%e 384: {1,1,1,1,1,1,1,2}

%e 480: {1,1,1,1,1,2,3}

%e 512: {1,1,1,1,1,1,1,1,1}

%e 768: {1,1,1,1,1,1,1,1,2}

%e 960: {1,1,1,1,1,1,2,3}

%t s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];

%t Select[Range[Length[s]],FreeQ[Take[s,#-1],s[[#]]]&]

%Y The unsorted version with multiplicity is A355386.

%Y This is the sorted version of A355391.

%Y A000005 counts divisors.

%Y A001221 counts prime indices without multiplicity.

%Y A001222 count prime indices with multiplicity.

%Y A070175 gives representatives for bigomega and omega, triangle A303555.

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

%K nonn

%O 1,2

%A _Gus Wiseman_, Jul 04 2022

%E a(41)-a(45) from _Amiram Eldar_, Jul 10 2022