A305149 -id:A305149 - cp's OEIS frontend

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 2, 2, 1, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 3, 1, 1, 2, 2, 2, 4, 1, 2, 2, 3, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 1, 4, 1, 2, 2, 1, 2, 3, 1, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 3, 1, 4, 1, 2, 1, 4, 2, 2, 2

Offset: 1

Gus Wiseman, Jul 29 2020

nonn

A number's prime signature (row n of A124010) is the sequence of positive exponents in its prime factorization, so a number has distinct prime multiplicities iff all the exponents in its prime signature are distinct.

			The a(n) sets for n = 36, 120, 144, 180 (ones not shown):
  {2,18}    {3,12,24}    {2,18,72}       {2,18}
  {3,12}    {5,20,40}    {3,9,18,72}     {3,12}
  {2,4,12}  {2,4,8,24}   {3,12,24,48}    {5,20}
  {3,9,18}  {2,4,8,40}   {3,12,24,72}    {5,45}
            {2,4,12,24}  {2,4,8,16,48}   {2,4,12}
            {2,4,20,40}  {2,4,8,24,48}   {2,4,20}
                         {2,4,8,24,72}   {3,9,18}
                         {2,4,12,24,48}  {3,9,45}
                         {2,4,12,24,72}

A336569 is the version for chains containing n.
A336571 is the non-maximal version.
A000005 counts divisors.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A032741 counts proper divisors.
A045778 counts strict factorizations.
A071625 counts distinct prime multiplicities.
A074206 counts strict chains of divisors from n to 1.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts chains of divisors.
A336422 counts divisible pairs of divisors, both in A130091.
A336424 counts factorizations using A130091.
A336500 counts divisors of n in A130091 with quotient also in A130091.
Cf. A002033, A005117, A098859, A118914, A124010, A305149, A327498, A327523, A336414, A336423, A336425, A336568.

strsigQ[n_]:=UnsameQ@@Last/@FactorInteger[n];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
strses[n_]:=If[n==1,{{}},Join@@Table[Append[#,d]&/@strses[d],{d,Select[Most[Divisors[n]],strsigQ]}]];
Table[Length[fasmax[strses[n]]],{n,100}]

A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 5, 3, 3, 1, 8, 1, 3, 3, 11, 1, 8, 1, 8, 3, 3, 1, 18, 3, 3, 5, 8, 1, 12, 1, 15, 3, 3, 3, 31, 1, 3, 3, 18, 1, 12, 1, 8, 8, 3, 1, 39, 3, 8, 3, 8, 1, 18, 3, 18, 3, 3, 1, 42, 1, 3, 8, 33, 3, 12, 1, 8, 3, 12, 1, 67, 1, 3, 8, 8, 3, 12, 1, 39, 11

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(12) = 8 pairs of factorizations:
  (2*2*3)|(2*2*3)
  (2*2*3)|(2*6)
  (2*2*3)|(3*4)
  (2*2*3)|(12)
    (2*6)|(12)
    (3*4)|(3*4)
    (3*4)|(12)
     (12)|(12)

Cf. A000688, A001055, A050336, A265947, A281113, A294068, A303386, A305149, A305150, A305193, A317144, A322435, A322439, A322440, A322441, A322442.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Tuples[facs[n],2],!Or@@divpropQ@@@Tuples[#]&]],{n,100}]

A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 4

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322437 at a(144) = 4, A322437(144) = 3.

First differs from A379958 at a(120) = 2, A379958(120) = 1.

			The a(240) = 5 pairs of factorizations::
  (4*4*15)|(4*6*10)
    (6*40)|(15*16)
    (8*30)|(12*20)
   (10*24)|(15*16)
   (12*20)|(15*16)

Antti Karttunen, Table of n, a(n) for n = 1..65537
Index entries for sequences computed from exponents in factorization of n.

Cf. A001055, A285572, A285573, A303362, A303386, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322437, A322442, A379958.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
divpropQ[x_,y_]:=And[x!=y,Divisible[x,y]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@divpropQ@@@Tuples[#],!Or@@divpropQ@@@Reverse/@Tuples[#]]&]],{n,100}]

factorizations(n, m=n, f=List([]), z=List([])) = if(1==n, listput(z,Vec(f)); z, my(newf); fordiv(n, d, if((d>1)&&(d<=m), newf = List(f); listput(newf,d); z = factorizations(n/d, d, newf, z))); (z));
is_proper_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if((fac1[i]!=fac2[j]) && (!(fac1[i]%fac2[j]) || !(fac2[j]%fac1[i])),return(0)))); (1); };
number_of_proper_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_proper_ndf_pair(z[i],z[j])));
A322438(n) = number_of_proper_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Jan 24 2025

Data section extended up to a(144) by Antti Karttunen, Jan 24 2025

A305253 Number of connected factorizations of n into factors greater than 1 whose distinct factors are pairwise indivisible.

Original entry on oeis.org

0, 1, 1, 2, 1, 1, 1, 2, 2, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 2, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 2, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 2, 3, 1, 1, 2, 1, 1, 1, 2, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 1, 1, 2, 1

Offset: 1

Gus Wiseman, May 28 2018

nonn

Given a finite multiset S of positive integers greater than one, let G(S) be the simple labeled graph with vertex set S and edges between any two vertices with a common divisor greater than 1. For example, G({6,14,15,35}) is a 4-cycle. This sequence counts factorizations S whose distinct factors are pairwise indivisible and such that G(S) is a connected graph.

			The a(360) = 8 factorizations: (360), (4*90), (10*36), (12*30), (15*24), (18*20), (4*6*15), (6*6*10).

Antti Karttunen, Table of n, a(n) for n = 1..20160
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305150, A305193.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
sacs[n_]:=Select[facs[n],Function[f,Length[zsm[f]]==1&&Select[Tuples[Union[f],2],UnsameQ@@#&&Divisible@@#&]=={}]]
Table[Length[sacs[n]],{n,500}]

is_connected(facs) = { my(siz=length(facs)); if(1==siz,1,my(m=matrix(siz,siz,i,j,(gcd(facs[i],facs[j])!=1))^siz); for(n=1,siz,if(0==vecmin(m[n,]),return(0))); (1)); };
A305253aux(n, m, facs) = if(1==n, is_connected(Vec(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&factorback(apply(x -> (x==d)||(x%d),Vec(facs))), newfacs = List(facs); listput(newfacs,d); s += A305253aux(n/d, d, newfacs))); (s));
A305253(n) = if(1==n,0,A305253aux(n, n, List([]))); \\ Antti Karttunen, Dec 06 2018

a(n) <= A305193(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Definition clarified by Gus Wiseman, more terms from Antti Karttunen, Dec 06 2018

A326082 Number of maximal sets of pairwise indivisible divisors of n.

Original entry on oeis.org

1, 2, 2, 3, 2, 3, 2, 4, 3, 3, 2, 5, 2, 3, 3, 5, 2, 5, 2, 5, 3, 3, 2, 8, 3, 3, 4, 5, 2, 7, 2, 6, 3, 3, 3, 9, 2, 3, 3, 8, 2, 7, 2, 5, 5, 3, 2, 12, 3, 5, 3, 5, 2, 8, 3, 8, 3, 3, 2, 15, 2, 3, 5, 7, 3, 7, 2, 5, 3, 7, 2, 15, 2, 3, 5, 5, 3, 7, 2, 12, 5, 3, 2, 15, 3

Offset: 1

Gus Wiseman, Jun 05 2019

nonn

Depends only on prime signature.

The non-maximal case is A096827.

			The maximal sets of pairwise indivisible divisors of n = 1, 2, 4, 8, 12, 24, 30, 32, 36, 48, 60 are:
   1   1   1   1   1     1      1         1    1       1       1
       2   2   2   12    24     30        2    36      48      60
           4   4   2,3   2,3    5,6       4    2,3     2,3     2,15
               8   3,4   3,4    2,15      8    2,9     3,4     3,20
                   4,6   3,8    3,10      16   3,4     3,8     4,30
                         4,6    2,3,5     32   4,18    4,6     5,12
                         6,8    6,10,15        9,12    6,8     2,3,5
                         8,12                  12,18   3,16    3,4,5
                                               4,6,9   6,16    4,5,6
                                                       8,12    3,4,10
                                                       12,16   6,15,20
                                                       16,24   10,12,15
                                                               12,15,20
                                                               12,20,30
                                                               4,6,10,15

Cf. A001055, A051026, A067992, A096827, A143824, A285572, A285573, A303362, A305148, A305149, A316476, A325861, A326023, A326077.

stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
fasmax[y_]:=Complement[y,Union@@(Most[Subsets[#]]&/@y)];
Table[Length[fasmax[Select[Rest[Subsets[Divisors[n]]],stableQ[#,Divisible]&]]],{n,100}]

A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 3, 2, 2, 1, 4, 1, 2, 2, 5, 1, 4, 1, 4, 2, 2, 1, 7, 2, 2, 3, 4, 1, 5, 1, 7, 2, 2, 2, 8, 1, 2, 2, 7, 1, 5, 1, 4, 4, 2, 1, 12, 2, 4, 2, 4, 1, 7, 2, 7, 2, 2, 1, 11, 1, 2, 4, 11, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 12, 5, 2, 1, 11, 2

Offset: 1

Gus Wiseman, May 28 2018

nonn

Given a factorization f consisting of positive integers greater than one, let G(F) be a multigraph with one vertex for each factor and n edges between any two vertices with n common divisors greater than 1. For example, G(6,14,15,35) is a 4-cycle; G(6,12) is a 2-cycle because 6 and 12 have multiple common divisors. This sequence counts factorizations f such that G(f) is a forest, meaning it has no cycles. [Comment edited by Robert Munafo, Mar 24 2024]

			The a(72) = 14 factorizations:
     (72)
    (2*36)     (3*24)    (4*18)    (8*9)
   (2*2*18)   (2*3*12)   (2*4*9)  (3*3*8) (3*4*6)
   (2*2*2*9)  (2*2*3*6) (2*3*3*4)
  (2*2*2*3*3)
not counted: (2*6*6) because 6 and 6 share multiple divisors; likewise (6*12) because 6 and 12 share multiple divisors.

Cf. A001970, A048143, A281116, A285572, A286518, A286520, A303386, A304714, A304716, A305149, A305193, A305253.

zsm[s_]:=With[{c=Select[Tuples[Range[Length[s]],2],And[Less@@#,GCD@@s[[#]]]>1&]},If[c=={},s,zsm[Union[Append[Delete[s,List/@c[[1]]],LCM@@s[[c[[1]]]]]]]]];
zensity[s_]:=Total[(PrimeNu[#]-1&)/@s]-PrimeNu[LCM@@s];
Table[Length[Select[facs[n],Function[f,And@@(zensity[Select[f,Function[x,Divisible[#,x]]]]==-1&/@zsm[f])]]],{n,200}]

Extensive clarification by Robert Munafo, Mar 22 2024

A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 6, 2, 2, 2, 4, 1, 5, 1, 3, 2, 2, 2, 9, 1, 2, 2, 6, 1, 5, 1, 4, 4, 2, 1, 9, 2, 4, 2, 4, 1, 6, 2, 6, 2, 2, 1, 11, 1, 2, 4, 4, 2, 5, 1, 4, 2, 5, 1, 14, 1, 2, 4, 4, 2, 5, 1, 9, 3, 2, 1, 11, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(72) = 14 factorizations:
  (2*2*2*3*3),
  (2*2*2*9), (2*2*3*6),
  (2*2*18), (2*3*12), (2*6*6), (3*3*8), (3*4*6),
  (2*36), (3*24), (4*18), (6*12), (8*9),
  (72).

Cf. A001055, A001597, A002865, A007916, A052410, A101417, A292886, A303707, A305149, A323053, A323087.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
Table[Length[Select[facs[n],stableQ[Union[#],IntegerQ[Log[#1,#2]]&]&]],{n,100}]

A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

Original entry on oeis.org

1, 1, 2, 1, 1, 3, 2, 1, 1, 2, 5, 1, 1, 1, 2, 3, 1, 1, 7, 2, 2, 1, 1, 1, 4, 1, 2, 2, 1, 2, 1, 1, 11, 1, 2, 1, 1, 4, 2, 1, 5, 1, 2, 1, 2, 2, 2, 1, 4, 1, 1, 1, 1, 1, 2, 2, 2, 3, 1, 15, 1, 7, 1, 1, 2, 2, 2, 1, 1, 4, 2, 1, 2, 1, 5, 1, 2, 1, 4, 2, 1, 1, 2, 1, 1, 1

Offset: 1

Gus Wiseman, Sep 15 2019

nonn

A prime index of n is a number m such that prime(m) divides n. The multiset of prime indices of n is row n of A112798. A number is stable if its distinct prime indices are pairwise indivisible. Stable numbers are listed in A316476.

			The a(26) = 4 factorizations of 45 into stable numbers:
  (3*3*5)
  (3*15)
  (5*9)
  (45)
The a(201) = 11 multiset partitions of the prime indices of 495 into stable multisets:
  {{2},{2},{3},{5}}
  {{2},{2},{3,5}}
  {{2},{3},{2,5}}
  {{2},{5},{2,3}}
  {{2},{2,3,5}}
  {{3},{2,2},{5}}
  {{3},{2,2,5}}
  {{2,2},{3,5}}
  {{5},{2,2,3}}
  {{2,3},{2,5}}
  {{2,2,3,5}}

Gus Wiseman, Sequences counting and encoding certain classes of multisets

See link for additional cross-references.

Cf. A001055, A303362, A305149.

nn=100;
facsusing[s_,n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsusing[Select[s,Divisible[n/d,#]&],n/d],Min@@#>=d&]],{d,Select[s,Divisible[n,#]&]}]];
stableQ[u_,Q_]:=!Apply[Or,Outer[#1=!=#2&&Q[#1,#2]&,u,u,1],{0,1}];
y=Select[Range[nn],stableQ[PrimePi/@First/@FactorInteger[#],Divisible]&];
Table[Length[facsusing[Rest[y],n]],{n,y}]

cp's OEIS Frontend

A336570 Number of maximal sets of proper divisors d|n, d < n, all belonging to A130091 (numbers with distinct prime multiplicities) and forming a divisibility chain.

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A322436 Number of pairs of factorizations of n into factors > 1 where no factor of the second properly divides any factor of the first.

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A322438 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one properly divides any factor of the other.

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A305253 Number of connected factorizations of n into factors greater than 1 whose distinct factors are pairwise indivisible.

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A326082 Number of maximal sets of pairwise indivisible divisors of n.

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A305254 Number of factorizations f of n into factors greater than 1 such that the graph of f is a forest.

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A323086 Number of factorizations of n into factors > 1 such that no factor is a power of any other (unequal) factor.

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A327520 Number of factorizations of the n-th stable number A316476(n) into stable numbers > 1.

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