A355386 -id:A355386 - cp's OEIS frontend

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 2, 2, 1, 1, 1, 2, 1, 1, 1, 3, 1, 1, 1

Offset: 1

Gus Wiseman, Jul 02 2022

nonn

The statistic omega = A001221 counts distinct prime factors (without multiplicity).
The statistic bigomega = A001222 counts prime factors with multiplicity.
If positive integers are regarded as arrows from the number of prime factors to the number of distinct prime factors, this sequence counts divisible composable pairs. Is there a nice choice of a composition operation making this into an associative category?

			The set of divisors of 180 satisfying the condition is {12, 18, 20, 30, 45}, so a(180) = 5.

The version with multiplicity is A181591.
For partitions we have A355383, with multiplicity A339006.
The version for compositions is A355384.
Positions of first appearances are A355386.
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, A133494, A181819, A182850, A323014, A323022, A323023, A355385, A355388.

Table[Length[Select[Divisors[n],PrimeOmega[#]==PrimeNu[n]&]],{n,100}]

A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

Original entry on oeis.org

1, 4, 8, 16, 32, 24, 128, 256, 512, 48, 2048, 4096, 8192, 16384, 96, 65536, 131072, 262144, 524288, 240, 192, 4194304, 8388608, 16777216, 33554432, 67108864, 134217728, 384, 536870912, 1073741824, 2147483648, 4294967296, 8589934592, 17179869184, 480, 768, 137438953472

Offset: 1

Gus Wiseman, Jul 04 2022

nonn

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 sequence is fully defined. Positions meeting this maximum are A185024, complement A006987.

			The terms together with their prime indices begin:
       1: {}
       4: {1,1}
       8: {1,1,1}
      16: {1,1,1,1}
      32: {1,1,1,1,1}
      24: {1,1,1,2}
     128: {1,1,1,1,1,1,1}
     256: {1,1,1,1,1,1,1,1}
     512: {1,1,1,1,1,1,1,1,1}
      48: {1,1,1,1,2}
    2048: {1,1,1,1,1,1,1,1,1,1,1}
    4096: {1,1,1,1,1,1,1,1,1,1,1,1}
    8192: {1,1,1,1,1,1,1,1,1,1,1,1,1}
   16384: {1,1,1,1,1,1,1,1,1,1,1,1,1,1}
      96: {1,1,1,1,1,2}
   65536: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  131072: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  262144: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
  524288: {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
     240: {1,1,1,1,2,3}
     192: {1,1,1,1,1,1,2}

Amiram Eldar, Table of n, a(n) for n = 1..168

Positions of powers of 2 are A185024, complement A006987.
Counting multiplicity gives A355386.
The sorted version is A355392.
A000005 counts divisors.
A001221 counts prime factors without multiplicity.
A001222 count prime factors with multiplicity.
A070175 gives representatives for bigomega and omega, triangle A303555.
Cf. A022811, A056239 , A071625, A118914, A181819, A323014, A323023, A355383 (with multiplicity A339006), A355384.

s=Table[Binomial[PrimeOmega[n],PrimeNu[n]],{n,1000}];
Table[Position[s,k][[1,1]],{k,Select[Union[s],SubsetQ[s,Range[#]]&]}]

f(n) = binomial(bigomega(n), omega(n)); \\ A181591
a(n) = my(k=1); while (f(k) != n, k++); k; \\ Michel Marcus, Jul 10 2022

binomial(bigomega(a(n)), omega(a(n))) = n.

a(22)-a(28) from Michel Marcus, Jul 10 2022

a(29)-a(37) from Amiram Eldar, Jul 10 2022

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

Original entry on oeis.org

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

Gus Wiseman, Jul 04 2022

nonn

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, ...

			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}

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

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.

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

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

cp's OEIS Frontend

A355382 Number of divisors d of n such that bigomega(d) = omega(n).

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A355391 Position of first appearance of n in A181591 = binomial(bigomega(n), omega(n)).

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

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