A305150 -id:A305150 - cp's OEIS frontend

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

1, 1, 2, 2, 3, 3, 4, 5, 6, 7, 9, 12, 12, 17, 20, 22, 28, 35, 39, 48, 55, 65, 79, 90, 105, 121, 143, 166, 190, 219, 254, 290, 332, 382, 436, 493, 567, 637, 729, 824, 931, 1052, 1186, 1334, 1504, 1691, 1894, 2123, 2380, 2664, 2968, 3319, 3704, 4119, 4586, 5110

Offset: 0

Gus Wiseman, May 26 2018

nonn

			The a(9) = 7 integer partitions are (9), (72), (54), (522), (333), (3222), (111111111).

Fausto A. C. Cariboni, Table of n, a(n) for n = 0..360 (terms 0..300 from Alois P. Heinz)

Cf. A000837, A001055, A001970, A007359, A007716, A051424, A078374, A100953, A285572, A285573, A302696, A303362, A305079, A305149, A305150.

Table[Length[Select[IntegerPartitions[n],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,20}]

More terms from Alois P. Heinz, May 26 2018

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 1, 3, 2, 1, 1, 3, 1, 1, 1, 5, 1, 3, 1, 3, 1, 1, 1, 5, 2, 1, 3, 3, 1, 1, 1, 7, 1, 1, 1, 6, 1, 1, 1, 5, 1, 1, 1, 3, 3, 1, 1, 9, 2, 3, 1, 3, 1, 5, 1, 5, 1, 1, 1, 4, 1, 1, 3, 11, 1, 1, 1, 3, 1, 1, 1, 11, 1, 1, 3, 3, 1, 1, 1, 9, 5, 1, 1, 4, 1, 1

Offset: 1

Gus Wiseman, Aug 03 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) factorizations for n = 2, 4, 8, 60, 16, 36, 32, 48:
  2  4    8      5*12     16       4*9      32         48
     2*2  2*4    3*20     4*4      3*12     4*8        4*12
          2*2*2  3*4*5    2*8      3*3*4    2*16       3*16
                 2*2*3*5  2*2*4    2*18     2*4*4      3*4*4
                          2*2*2*2  2*2*9    2*2*8      2*24
                                   2*2*3*3  2*2*2*4    2*3*8
                                            2*2*2*2*2  2*2*12
                                                       2*2*3*4
                                                       2*2*2*2*3

A327523 is the case when n is restricted to belong to A130091 also.
A001055 counts factorizations.
A007425 counts divisors of divisors.
A045778 counts strict factorizations.
A074206 counts ordered factorizations.
A130091 lists numbers with distinct prime multiplicities.
A181796 counts divisors with distinct prime multiplicities.
A253249 counts nonempty chains of divisors.
A281116 counts factorizations with no common divisor.
A302696 lists numbers whose prime indices are pairwise coprime.
A305149 counts stable factorizations.
A320439 counts factorizations using A289509.
A327498 gives the maximum divisor with distinct prime multiplicities.
A336500 counts divisors of n in A130091 with quotient also in A130091.
A336568 = not a product of two numbers with distinct prime multiplicities.
A336569 counts maximal chains of elements of A130091.
A337256 counts chains of divisors.
Cf. A071625, A080688, A098859, A118914, A124010, A167865, A294068, A303707, A305150, A322453, A336420, A336570, A336571.

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,#]&]}]];
Table[Length[facsusing[Select[Range[2,n],UnsameQ@@Last/@FactorInteger[#]&],n]],{n,100}]

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 2, 3, 1, 3, 1, 3, 2, 2, 1, 4, 2, 2, 2, 3, 1, 5, 1, 2, 2, 2, 2, 6, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 2, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 8, 1, 2, 3, 4, 2, 5, 1, 3, 2, 5, 1, 6, 1, 2, 3, 3, 2, 5, 1, 5, 3, 2, 1, 8, 2, 2, 2, 4, 1, 8, 2, 3, 2, 2, 2, 6, 1, 3, 3, 6, 1, 5, 1, 4, 5

Offset: 1

Gus Wiseman, May 26 2018

nonn

			The a(60) = 8 factorizations are (2*2*3*5), (2*2*15), (3*4*5), (3*20), (4*15), (5*12), (6*10), (60). Missing from this list are (2*3*10), (2*5*6), (2*30).

Antti Karttunen, Table of n, a(n) for n = 1..10000
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000

Cf. A001055, A001970, A007716, A034444, A045778, A259936, A281116, A285572, A302242, A303386, A303431, A305001, A305148, A305150.

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],Select[Tuples[Union[#],2],UnsameQ@@#&&Divisible@@#&]=={}&]],{n,100}]

pairwise_indivisible(v) = { for(i=1,#v,for(j=i+1,#v,if(!(v[j]%v[i]),return(0)))); (1); };
A305149(n, m=n, facs=List([])) = if(1==n, pairwise_indivisible(Set(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A305149(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Oct 08 2018

More terms from Antti Karttunen, Oct 08 2018

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Original entry on oeis.org

1, 0, 0, 1, 0, 1, 0, 2, 1, 1, 0, 3, 0, 1, 1, 5, 0, 3, 0, 3, 1, 1, 0, 7, 1, 1, 2, 3, 0, 4, 0, 7, 1, 1, 1, 15, 0, 1, 1, 7, 0, 4, 0, 3, 3, 1, 0, 16, 1, 3, 1, 3, 0, 7, 1, 7, 1, 1, 0, 18, 0, 1, 3, 16, 1, 4, 0, 3, 1, 4, 0, 32, 0, 1, 3, 3, 1, 4, 0, 16, 5, 1, 0, 18, 1

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

			The a(36) = 15 pairs of factorizations:
  (2*2*3*3)|(4*9)
  (2*2*3*3)|(6*6)
  (2*2*3*3)|(36)
    (2*2*9)|(6*6)
    (2*2*9)|(36)
    (2*3*6)|(4*9)
    (2*3*6)|(36)
     (2*18)|(36)
    (3*3*4)|(6*6)
    (3*3*4)|(36)
     (3*12)|(36)
      (4*9)|(6*6)
      (4*9)|(36)
      (6*6)|(4*9)
      (6*6)|(36)

Cf. A000688, A001055, A050336, A265947, A294068, A305149, A305150, A305193, A317144, A322436, A322437, A322438, A322439, A322440, A322441, A322442.

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

A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one 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

Offset: 1

Gus Wiseman, Dec 08 2018

nonn

First differs from A322438 at a(144) = 3, A322438(144) = 4.
From Antti Karttunen, Dec 11 2020: (Start)
Zeros occur on numbers that are either of the form p^k, or q * p^k, or p*q*r, for some primes p, q, r, and exponent k >= 0. [Note also that in all these cases, when x > 1, A307408(x) = 2+A307409(x) = 2 + (A001222(x) - 1)*A001221(x) = A000005(x)].
Proof:
  It is easy to see that for such numbers it is not possible to obtain two such distinct factorizations, that no factor of the other would not divide some factor of the other.
  Conversely, the complement set of above is formed of such composites n that have at least one unitary divisor that is either of the form
  (1) p^x * q^y, with x, y >= 2,
  or
  (2) p^x * q^y * r^z, with x >= 2, and y, z >= 1,
  or
  (3) p^x * q^y * r^z * s^w, with x, y, z, w >= 1,
  where p, q, r, s are distinct primes. Let's indicate with C the remaining portion of k coprime to p, q, r and s (which could be also 1). Then in case (1) we can construct two factorizations, the first having factors (p*q*C) and (p^(x-1) * q^(y-1)), and the second having factors (p^x * C) and (q^y) that are guaranteed to satisfy the condition that no factor in the other factorization divides any of the factors of the other factorization. For case (2) pairs like {(p * q^y * C), (p^(x-1) * r^z)} and {(p^x * C), (q^y * r^z)}, and for case (3) pairs like {(p^x * q^y * C), (r^z * s^w)} and {(p^x * r^z * C), (q^y * s^w)} offer similar examples, therefore a(n) > 0 for all such cases.
(End)

			The a(120) = 2 pairs of such factorizations:
   (6*20)|(8*15)
   (8*15)|(10*12)
The a(144) = 3 pairs of factorizations:
   (6*24)|(9,16)
   (8*18)|(12*12)
   (9*16)|(12*12)
The a(210) = 3 pairs of factorizations:
   (6*35)|(10*21)
   (6*35)|(14*15)
  (10*21)|(14*15)
[Note that 210 is the first squarefree number obtaining nonzero value]
The a(240) = 4 pairs of factorizations:
   (6*40)|(15*16)
   (8*30)|(12*20)
  (10*24)|(15*16)
  (12*20)|(15*16)
The a(1728) = 14 pairs of factorizations:
    (6*6*48)|(27*64)
   (6*12*24)|(27*64)
     (6*288)|(27*64)
    (8*8*27)|(12*12*12)
  (12*12*12)|(27*64)
  (12*12*12)|(32*54)
    (12*144)|(27*64)
    (12*144)|(32*54)
    (16*108)|(24*72)
     (18*96)|(27*64)
     (24*72)|(27*64)
     (24*72)|(32*54)
     (27*64)|(36*48)
     (32*54)|(36*48)

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

Cf. A000688, A001055, A285572, A285573, A294068, A303362, A304713, A305149, A305150, A305193, A316476, A317144, A322435, A322436, A322441, A322442, A339626.

Cf. also comments A307408 and A307409.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Table[Length[Select[Subsets[facs[n],{2}],And[!Or@@Divisible@@@Tuples[#],!Or@@Divisible@@@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_ndf_pair(fac1,fac2) = { for(i=1,#fac1,for(j=1,#fac2,if(!(fac1[i]%fac2[j])||!(fac2[j]%fac1[i]),return(0)))); (1); };
number_of_ndfpairs(z) = sum(i=1,#z,sum(j=i+1,#z,is_ndf_pair(z[i],z[j])));
A322437(n) = number_of_ndfpairs(Vec(factorizations(n))); \\ Antti Karttunen, Dec 10 2020

For n > 0, a(A002110(n)) = A322441(n)/2 = A339626(n). - Antti Karttunen, Dec 10 2020

Data section extended up to a(120) and more examples added by Antti Karttunen, Dec 10 2020

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

A323087 Number of strict factorizations of n into factors > 1 such that no factor is a power of any other factor.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 2, 1, 3, 1, 2, 2, 1, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 1, 3, 1, 5, 1, 2, 2, 2, 2, 5, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 9, 1, 2, 3, 1, 2, 5, 1, 3, 2, 5, 1, 8, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

			The a(60) = 9 factorizations:
  (2*3*10), (2*5*6), (3*4*5),
  (2*30), (3*20), (4*15), (5*12), (6*10),
  (60).

Cf. A001597, A007916, A025147, A045778, A052410, A087897, A275972, A305150, A323054, A323086, A323090, A323091.

strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[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[strfacs[n],stableQ[#,IntegerQ[Log[#1,#2]]&]&]],{n,100}]

A323091 Number of strict knapsack factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 5, 1, 2, 2, 3, 1, 5, 1, 3, 2, 2, 2, 4, 1, 2, 2, 5, 1, 5, 1, 3, 3, 2, 1, 7, 1, 3, 2, 3, 1, 5, 2, 5, 2, 2, 1, 9, 1, 2, 3, 3, 2, 5, 1, 3, 2, 5, 1, 9, 1, 2, 3, 3, 2, 5, 1, 7, 2, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Jan 04 2019

nonn

A strict knapsack factorization is a finite set of positive integers > 1 such that every subset has a different product.

			The a(144) = 11 factorizations:
  (144),
  (2*72), (3*48), (4*36),(6*24), (8*18), (9*16),
  (2*3*24), (2*4*18), (2*8*9), (3*6*8).
Missing from this list are (2*6*12), (3*4*12), (2*3*4*6), which are not knapsack.

Cf. A001055, A045778, A108917, A120641, A275972, A292886, A305150, A323087.

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

a(prime^n) = A275972(n).

cp's OEIS Frontend

A305148 Number of integer partitions of n whose distinct parts are pairwise indivisible.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

Extensions

A336424 Number of factorizations of n where each factor belongs to A130091 (numbers with distinct prime multiplicities).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A305149 Number of factorizations of n whose distinct factors are pairwise indivisible and greater than 1.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A322435 Number of pairs of factorizations of n into factors > 1 where no factor of the second divides any factor of the first.

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

A322437 Number of unordered pairs of factorizations of n into factors > 1 where no factor of one divides any factor of the other.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Examples

Crossrefs

Programs

Mathematica

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs