A325990 -id:A325990 - cp's OEIS frontend

A325988 Number of covering (or complete) factorizations of n.

1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 2, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1

Offset: 1

Gus Wiseman, May 30 2019

nonn

First differs from A072911 at a(64) = 5, A072911(64) = 4.

A covering factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of some submultiset of the factors.

			The a(64) = 5 factorizations:
  (2*2*2*2*2*2)
  (2*2*2*2*4)
  (2*2*2*8)
  (2*2*4*4)
  (2*4*8)
The a(96) = 4 factorizations:
  (2*2*2*2*2*3)
  (2*2*2*3*4)
  (2*2*3*8)
  (2*3*4*4)

Cf. A002033, A103295, A126796, A188431.

Cf. A325684, A325781, A325786, A325788, A325986, A325989, A325990.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Union[Times@@@Subsets[#]]==Divisors[n]&]],{n,100}]

a(2^n) = A126796(n).

A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 216, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Sep 11 2024

Subsequence of A060476 and differs from it by not having the terms 1, 256, 768, 1280, 1792, 2304, ... .
Subsequence of A295661 and first differs from it at n = 51: A295661(51) = 432 is not a term of this sequence.
First differs from A325990 at n = 30: A325990(30) = 256 is not a term of this sequence.
Nonsquarefree numbers k such that A051903(k) is odd, or equivalently, numbers k such that A051903(k) is an odd number that is larger than 1.
The asymptotic density of this sequence is Sum_{k>=3} (-1)^(k+1) * (1 - 1/zeta(k)) = 0.11615617754774636364... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Complement of A368714 within A013929.

Cf. A051903, A060476, A295661, A325990.

q[n_] := n > 1 && OddQ[n]; Select[Range[500], q[Max[FactorInteger[#][[;; , 2]]]] &]

is(k) = k > 1 && apply(x -> (x > 1 && x % 2), vecmax(factor(k)[, 2]));

A325989 Number of perfect factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, May 30 2019

nonn

A perfect factorization of n is an orderless factorization of n into factors > 1 such that every divisor of n is the product of exactly one submultiset of the factors. This is the intersection of covering (or complete) factorizations (A325988) and knapsack factorizations (A292886).

			The a(216) = 4 perfect factorizations:
  (2*2*2*3*3*3)
  (2*2*2*3*9)
  (2*3*3*3*4)
  (2*3*4*9)

Positions of terms > 1 are A325990.

Cf. A002033, A103300, A292886, A325685, A325780, A325787, A325789, A325988.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[facs[n],Sort[Times@@@Union[Subsets[#]]]==Divisors[n]&]],{n,100}]

a(2^n) = A002033(n).

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Original entry on oeis.org

8, 24, 27, 32, 40, 54, 56, 72, 88, 96, 104, 108, 120, 125, 128, 135, 136, 152, 160, 168, 184, 189, 200, 224, 232, 243, 248, 250, 264, 270, 280, 288, 296, 297, 312, 328, 343, 344, 351, 352, 360, 375, 376, 378, 384, 392, 408, 416, 424, 432, 440, 456, 459, 472, 480, 486, 488, 500

Offset: 1

Amiram Eldar, Nov 09 2024

First differs from A295661, A325990 and A376142 at n = 24: A295661(24) = A325990(24) = A376142(24) = 216 = 2^3 * 3^3 is not a term of this sequence.
Differs from A060476 by having the terms 432, 648, 1728, ..., and not having the terms 1, 216, 256, 768, 864, ... .
The asymptotic density of this sequence is Product_{p prime} (1 - 1/(p^2*(p+1))) * Sum_{p prime} (1/(p^3+p^2-1)) = 0.11498368544519741081... .

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A295661.
Subsequences: A065036, A143610, A163569.
Cf. A060476, A065465, A325990, A376142, A377845.

q[n_] := Count[FactorInteger[n][[;; , 2]], _?(# > 1 && OddQ[#] &)] == 1; Select[Range[500], q]

is(k) = #select(x -> x>1 && x%2, factor(k)[, 2]) == 1;

cp's OEIS Frontend

A325988 Number of covering (or complete) factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A376142 Nonsquarefree numbers whose prime factorization has a maximum exponent that is odd.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A325989 Number of perfect factorizations of n.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A377844 Numbers that have a single odd exponent larger than 1 in their prime factorization.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI