A281116 -id:A281116 - cp's OEIS frontend

A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

6, 8, 10, 14, 15, 16, 21, 22, 26, 27, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 64, 65, 69, 74, 77, 81, 82, 85, 86, 87, 91, 93, 94, 95, 96, 106, 111, 115, 118, 119, 120, 122, 123, 125, 129, 133, 134, 141, 142, 143, 144, 145, 146, 155, 158, 159, 160, 161, 166

Offset: 1

Gus Wiseman, Sep 15 2018

nonn

From Tian Vlasic, Jan 01 2022: (Start)
Numbers that have an equal number of even- and odd-length unordered factorizations into distinct factors.
For prime p, by the pentagonal number theorem, p^k is a term if and only if k is in A090864.
For primes p and q, p*q^k is a term if and only if k = A000326(m)+N with 0 <= N < m. (End)

			16 = 2*8 = 4*4 = 2*2*4 = 2*2*2*2 has an equal number of even-length factorizations and odd-length factorizations into distinct factors (1). - _Tian Vlasic_, Dec 31 2021

Leonhard Euler, On the remarkable properties of the pentagonal numbers, arXiv:math/0505373 [math.HO], 2005.
Leonhard Euler, De mirabilibus proprietatibus numerorum pentagonalium, par. 2.
Eric Weisstein's World of Mathematics, Pentagonal Number Theorem
Wikipedia, Pentagonal number theorem

Complement of A319237.

Cf. A001055, A045778, A114592, A162247, A190938, A281116, A281118, A303386, A316441, A319240.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Join@@Position[Table[Sum[(-1)^Length[f],{f,Select[facs[n],UnsameQ@@#&]}],{n,100}],0]

A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

Original entry on oeis.org

1, 0, 1, 3, 10, 25, 72, 182, 502, 1332, 3720, 10380, 30142, 88842, 270569, 842957, 2703060, 8885029, 29990388, 103743388, 367811233, 1334925589, 4957151327, 18817501736, 72972267232, 288863499000, 1166486601571, 4802115258807, 20141268290050, 86017885573548, 373852868791639

Offset: 0

Gus Wiseman, Sep 27 2018

nonn

The weight of a set multipartition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(2) = 1 through a(4) = 10 set multipartitions:
  {{1},{2}}   {{1},{2,3}}     {{1},{2,3,4}}
             {{1},{2},{2}}    {{1,2},{3,4}}
             {{1},{2},{3}}   {{1},{1},{2,3}}
                             {{1},{2},{1,2}}
                             {{1},{2},{3,4}}
                             {{1},{3},{2,3}}
                            {{1},{1},{2},{2}}
                            {{1},{2},{2},{2}}
                            {{1},{2},{3},{3}}
                            {{1},{2},{3},{4}}

Andrew Howroyd, Table of n, a(n) for n = 0..50

Cf. A007716, A049311, A281116, A283877, A316980, A317752, A317755, A317757, A319616.

Cf. A319077, A319751, A319755, A319778, A319781, A319791.

WeighT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, (-1)^(n-1)/n))))-1, -#v)}
permcount(v) = {my(m=1, s=0, k=0, t); for(i=1, #v, t=v[i]; k=if(i>1&&t==v[i-1], k+1, 1); m*=t*k; s+=t); s!/m}
K(q, t, k)={WeighT(Vec(sum(j=1, #q, gcd(t, q[j])*x^lcm(t, q[j])) + O(x*x^k), -k))}
R(q, n)={vector(n, t, x*Ser(K(q, t, n)/t))}
a(n)={if(n==0, 1, my(s=0); forpart(q=n, my(u=R(q,n)); s+=permcount(q)*polcoef(exp(sum(t=1, n, u[t], O(x*x^n))) - exp(sum(t=1, n\2, x^t*u[t], O(x*x^n)))/(1-x), n)); s/n!)} \\ Andrew Howroyd, May 30 2023

Terms a(11) and beyond from Andrew Howroyd, May 30 2023

A319767 Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.

Original entry on oeis.org

1, 1, 1, 5, 73

Offset: 0

Gus Wiseman, Sep 27 2018

The dual of a multiset partition has, for each vertex, one part consisting of the indices (or positions) of the parts containing that vertex, counted with multiplicity. For example, the dual of {{1,2},{2,2}} is {{1},{1,2,2}}.
A multiset partition is intersecting iff no two parts are disjoint. The dual of a multiset partition is intersecting iff every pair of distinct vertices appear together in some part.
The weight of a multiset partition is the sum of sizes of its parts. Weight is generally not the same as number of vertices.

			Non-isomorphic representatives of the a(1) = 1 through a(3) = 5 multiset partitions:
1: {{1}}
2: {{2},{1,2}}
3: {{3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3}}
   {{1,3},{2,3},{1,2,3}}
   {{3},{1,3},{2,3},{1,2,3}}
   {{1,2},{1,3},{2,3},{1,2,3}}

Cf. A007716, A281116, A283877, A305854, A306006, A316980, A316983, A317757, A319616.

Cf. A319752, A319765, A319766, A319768, A319769, A319773, A319774.

A319786 Number of factorizations of n where no two factors are relatively prime.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Sep 27 2018

nonn

First differs from A305193 at a(36) = 4, A305193(36) = 5.

a(n) depends only on prime signature of n (cf. A025487). - Antti Karttunen, Nov 07 2018

			The a(48) = 7 factorizations are (2*2*2*6), (2*2*12), (2*4*6), (2*24), (4*12), (6*8), (48).

Antti Karttunen, Table of n, a(n) for n = 1..11520
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..100000
Index entries for sequences computed from exponents in factorization of n

Cf. A001055, A007716, A045778, A049311, A162247, A281116, A283877, A305193, A305854, A306006, A316980, A318749.

Cf. A319755, A319759, A319760, A319765, A319779, A319787, A319782, A319789.

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],!Or@@CoprimeQ@@@Subsets[#,{2}]&]],{n,100}]

A319786(n, m=n, facs=List([])) = if(1==n, (1!=gcd(Vec(facs))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A319786(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Nov 07 2018

More terms from Antti Karttunen, Nov 07 2018

A066637 Total number of elements in all factorizations of n with all factors > 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 3, 1, 6, 3, 3, 1, 8, 1, 3, 3, 12, 1, 8, 1, 8, 3, 3, 1, 17, 3, 3, 6, 8, 1, 10, 1, 20, 3, 3, 3, 22, 1, 3, 3, 17, 1, 10, 1, 8, 8, 3, 1, 34, 3, 8, 3, 8, 1, 17, 3, 17, 3, 3, 1, 27, 1, 3, 8, 35, 3, 10, 1, 8, 3, 10, 1, 46, 1, 3, 8, 8, 3, 10, 1, 34, 12, 3, 1, 27, 3, 3, 3, 17, 1, 27, 3, 8, 3, 3, 3

Offset: 1

Amarnath Murthy, Dec 28 2001

nonn

From Gus Wiseman, Apr 18 2021: (Start)
Number of ways to choose a factor index or position in a factorization of n. The version selecting a factor value is A339564. For example, the factorizations of n = 2, 4, 8, 12, 16, 24, 30 with a selected position (in parentheses) are:
  ((2))  ((4))    ((8))      ((12))     ((16))       ((24))       ((30))
         ((2)*2)  ((2)*4)    ((2)*6)    ((2)*8)      ((3)*8)      ((5)*6)
         (2*(2))  (2*(4))    (2*(6))    (2*(8))      (3*(8))      (5*(6))
                  ((2)*2*2)  ((3)*4)    ((4)*4)      ((4)*6)      ((2)*15)
                  (2*(2)*2)  (3*(4))    (4*(4))      (4*(6))      (2*(15))
                  (2*2*(2))  ((2)*2*3)  ((2)*2*4)    ((2)*12)     ((3)*10)
                             (2*(2)*3)  (2*(2)*4)    (2*(12))     (3*(10))
                             (2*2*(3))  (2*2*(4))    ((2)*2*6)    ((2)*3*5)
                                        ((2)*2*2*2)  (2*(2)*6)    (2*(3)*5)
                                        (2*(2)*2*2)  (2*2*(6))    (2*3*(5))
                                        (2*2*(2)*2)  ((2)*3*4)
                                        (2*2*2*(2))  (2*(3)*4)
                                                     (2*3*(4))
                                                     ((2)*2*2*3)
                                                     (2*(2)*2*3)
                                                     (2*2*(2)*3)
                                                     (2*2*2*(3))
(End)

			a(12) = 8: there are 4 factorizations of 12: (12), (6*2), (4*3), (3*2*2) having 1, 2, 2, 3 elements respectively, a total of 8.

Amarnath Murthy, Generalization of Partition function, Introducing Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.
Amarnath Murthy, Length and extent of Smarandache Factor partitions, Smarandache Notions Journal, Vol. 11, 1-2-3, Spring 2000.

Alois P. Heinz, Table of n, a(n) for n = 1..20000 (first 1000 terms from T. D. Noe)

The version for normal multisets is A001787.
The version for compositions is A001792.
The version for partitions is A006128 (strict: A015723).
Choosing a value instead of position gives A339564.
A000070 counts partitions with a selected part.
A001055 counts factorizations.
A002033 and A074206 count ordered factorizations.
A067824 counts strict chains of divisors starting with n.
A336875 counts compositions with a selected part.
Cf. A000005, A000041, A045778, A050336, A066186, A162247, A264401, A281116, A292504, A292886, A322794.

# Return a list of lists which are factorizations (product representations)
# of n. Within each sublist, the factors are sorted. A minimum factor in
# each element of sublists returned can be specified with 'mincomp'.
# If mincomp=2, the number of sublists contained in the list returned is A001055(n).
# Example:
# n=8 and mincomp=2 return [[2,2,2],[4,8],[8]]
listProdRep := proc(n,mincomp)
    local dvs,resul,f,i,j,rli,tmp ;
    resul := [] ;
    # list returned is empty if n < mincomp
    if n >= mincomp then
        if n = 1 then
            RETURN([1]) ;
        else
            # compute the divisors, and take each divisor
            # as a head element (minimum element) of one of the
            # sublists. Example: for n=8 use {1,2,4,8}, and consider
            # (for mincomp=2) sublists [2,...], [4,...] and [8].
            dvs := numtheory[divisors](n) ;
            for i from 1 to nops(dvs) do
                # select the head element 'f' from the divisors
                f := op(i,dvs) ;
                # if this is already the maximum divisor n
                # itself, this head element is the last in
                # the sublist
                if f =n and f >= mincomp then
                    resul := [op(resul),[f]] ;
                elif f >= mincomp then
                    # if this is not the maximum element
                    # n itself, produce all factorizations
                    # of the remaining factor recursively.
                    rli := procname(n/f,f) ;
                    # Prepend all the results produced
                    # from the recursion with the head
                    # element for the result.
                    for j from 1 to nops(rli) do
                        tmp := [f,op(op(j,rli))] ;
                        resul := [op(resul),tmp] ;
                    od ;
                fi ;
            od ;
        fi ;
    fi ;
    resul ;
end:
A066637 := proc(n)
    local f,d;
    a := 0 ;
    for d in listProdRep(n,2) do
        a := a+nops(d) ;
    end do:
    a ;
end proc: # R. J. Mathar, Jul 11 2013
# second Maple program:
with(numtheory):
b:= proc(n, k) option remember; `if`(n>k, 0, [1$2])+
      `if`(isprime(n), 0, (p-> p+[0, p[1]])(add(
      `if`(d>k, 0, b(n/d, d)), d=divisors(n) minus {1, n})))
    end:
a:= n-> `if`(n<2, 0, b(n$2)[2]):
seq(a(n), n=1..120); # Alois P. Heinz, Feb 12 2019

g[1, r_] := g[1, r]={1, 0}; g[n_, r_] := g[n, r]=Module[{ds, i, val}, ds=Select[Divisors[n], 1<#<=r&]; val={0, 0}+Sum[g[n/ds[[i]], ds[[i]]], {i, 1, Length[ds]}]; val+{0, val[[1]]}]; a[n_] := g[n, n][[2]]; a/@Range[95] (* g[n, r] = {c, f}, where c is the number of factorizations of n with factors <= r and f is the total number of factors in them. - Dean Hickerson, Oct 28 2002 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];Table[Sum[Length[fac],{fac,facs[n]}],{n,50}] (* Gus Wiseman, Apr 18 2021 *)

A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

Original entry on oeis.org

0, 1, 1, 3, 1, 4, 1, 7, 3, 4, 1, 12, 1, 4, 4, 14, 1, 12, 1, 12, 4, 4, 1, 29, 3, 4, 7, 12, 1, 17, 1, 27, 4, 4, 4, 36, 1, 4, 4, 29, 1, 17, 1, 12, 12, 4, 1, 62, 3, 12, 4, 12, 1, 29, 4, 29, 4, 4, 1, 53, 1, 4, 12, 47, 4, 17, 1, 12, 4, 17, 1, 90, 1, 4, 12, 12, 4, 17

Offset: 1

Gus Wiseman, Mar 27 2018

nonn

For a finite multiset p of positive integers greater than 1 with product n, a pair (t > 1, p) is defined to be a subset-product if there exists a nonempty submultiset of p with product t.

			The a(12) = 12 subset-products:
12<=(2*2*3), 6<=(2*2*3), 4<=(2*2*3), 3<=(2*2*3), 2<=(2*2*3),
12<=(2*6),   6<=(2*6),   4<=(3*4),   3<=(3*4),   2<=(2*6),
12<=(3*4),
12<=(12).
The a(16) = 14 subset-products:
16<=(16),
16<=(4*4),
16<=(2*8),     8<=(2*8),     4<=(4*4),     2<=(2*8),
16<=(2*2*4),   8<=(2*2*4),   4<=(2*2*4),   2<=(2*2*4),
16<=(2*2*2*2), 8<=(2*2*2*2), 4<=(2*2*2*2), 2<=(2*2*2*2).

Cf. A001055, A000712, A045778, A108917, A162247, A276024, A281116, A284640, A292886, A301829, A301830.

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

A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Apr 29 2018

nonn

A periodic factorization of n is a finite multiset of positive integers greater than 1 whose product is n and whose multiplicities have a common divisor greater than 1. Note that a factorization of a number that is not a perfect power (A007916) is always aperiodic (A303386), so the indices of nonzero entries of this sequence all lie at perfect powers (A001597).

			The a(900) = 5 periodic factorizations are (2*2*3*3*5*5), (2*2*15*15), (3*3*10*10), (5*5*6*6), (30*30).

Antti Karttunen, Table of n, a(n) for n = 1..100000

Cf. A000740, A000837, A001055, A001597, A007716, A007916, A052409, A052410, A281116, A303547, A303552, A303553, A303386, A303707, A303708, A303710.

radQ[n_]:=Or[n===1,GCD@@FactorInteger[n][[All,2]]===1];
facsr[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facsr[n/d],Min@@#>=d&]],{d,Select[Rest[Divisors[n]],radQ]}]];
Table[Length[Select[facsr[n],GCD@@Length/@Split[#]!=1&]],{n,200}]

gcd_of_multiplicities(lista) = { my(u=length(lista)); if(u<2, u, my(g=0, pe = lista[1], j=1); for(i=2,u,if(lista[i]==pe, j++, g = gcd(j,g); j=1; pe = lista[i])); gcd(g,j)); }; \\ the supplied lista (newfacs) should be monotonic
A303709(n, m=n, facs=List([])) = if(1==n, (1!=gcd_of_multiplicities(facs)), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m)&&!ispower(d), newfacs = List(facs); listput(newfacs,d); s += A303709(n/d, d, newfacs))); (s)); \\ Antti Karttunen, Dec 06 2018

a(n) <= A303553(n) <= A001055(n). - Antti Karttunen, Dec 06 2018

Changed a(1) to 1 by Gus Wiseman, Dec 06 2018

A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Aug 06 2018

			Triangle begins:
   1:  0
   2:  0  1
   3:  0  1
   4:  0  1  1
   5:  0  1
   6:  1  0  0  1
   7:  0  1
   8:  0  2  0  1
   9:  0  1  1
  10:  1  0  0  1
  11:  0  1
  12:  2  1  0  0  0  1
  13:  0  1
  14:  1  0  0  1
  15:  1  0  0  1
  16:  0  3  1  0  1
  17:  0  1
  18:  2  0  1  0  0  1
  19:  0  1
  20:  2  1  0  0  0  1

Row lengths are A000005. Row sums are A001055. First column is A281116. Number of nonzero terms in each row is A317751.
Cf. A007562, A007716, A014963, A027750, A045778, A050370, A162247, A289509, A303386, A303546.
Cf. A317752, A317755, A317757.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
goc[n_,m_]:=Length[Select[facs[n],And[And@@(Divisible[#,m]&/@#),GCD@@(#/m)==1]&]];
Table[goc[n,d],{n,30},{d,Divisors[n]}]

Name edited by Peter Munn, Mar 05 2025

A317775 Number of strict multiset partitions of strongly normal multisets of size n, where a multiset is strongly normal if it spans an initial interval of positive integers with weakly decreasing multiplicities.

Original entry on oeis.org

1, 3, 10, 36, 136, 596, 2656, 13187, 68226, 381572, 2233091, 13940407, 90981030, 626911429, 4509031955, 33987610040, 266668955183, 2180991690286, 18512572760155, 163103174973092, 1487228204311039, 14027782824491946, 136585814043190619, 1371822048393658001, 14190528438090988629

Offset: 1

Gus Wiseman, Aug 06 2018

nonn

			The a(3) = 10 strict multiset partitions:
  {{1,1,1}}, {{1},{1,1}},
  {{1,1,2}}, {{1},{1,2}}, {{2},{1,1}},
  {{1,2,3}}, {{1},{2,3}}, {{2},{1,3}}, {{3},{1,2}}, {{1},{2},{3}}.

Andrew Howroyd, Table of n, a(n) for n = 1..50

Cf. A001055, A007716, A045778, A281116, A317449, A317584, A317654, A317755, A317757, A317776.

sps[{}]:={{}};sps[set:{i_,_}]:=Join@@Function[s,Prepend[#,s]&/@sps[Complement[set,s]]]/@Cases[Subsets[set],{i,_}];
mps[set_]:=Union[Sort[Sort/@(#/.x_Integer:>set[[x]])]&/@sps[Range[Length[set]]]];
strnorm[n_]:=Flatten[MapIndexed[Table[#2,{#1}]&,#]]&/@IntegerPartitions[n];
Table[Length[Select[Join@@mps/@strnorm[n],UnsameQ@@#&]],{n,6}]

EulerT(v)={Vec(exp(x*Ser(dirmul(v, vector(#v, n, 1/n))))-1, -#v)}
D(p, n)={my(v=vector(n)); for(i=1, #p, v[p[i]]++); my(u=EulerT(v)); Vec(1/prod(k=1, n, 1 - u[k]*x^k + O(x*x^n))-1,-n)/prod(i=1, #v, i^v[i]*v[i]!)}
seq(n)={my(s); for(k=1, n, forpart(p=k, s+=(-1)^(k+#p)*D(p,n))); s[n]+=1; s/2} \\ Andrew Howroyd, Dec 30 2020

Terms a(10) and beyond from Andrew Howroyd, Dec 30 2020

A318720 Numbers k such that there exists a strict relatively prime factorization of k in which no pair of factors is relatively prime.

Original entry on oeis.org

900, 1764, 1800, 2700, 3528, 3600, 4356, 4500, 4900, 5292, 5400, 6084, 6300, 7056, 7200, 8100, 8712, 8820, 9000, 9800, 9900, 10404, 10584, 10800, 11025, 11700, 12100, 12168, 12348, 12600, 12996, 13068, 13500, 14112, 14400, 14700, 15300, 15876, 16200, 16900

Offset: 1

Gus Wiseman, Sep 02 2018

nonn

From Amiram Eldar, Nov 01 2020: (Start)
Also, numbers with more than two non-unitary prime divisors, i.e., numbers k such that A056170(k) > 2, or equivalently, numbers divisible by the squares of three distinct primes.
The complement of the union of A005117, A190641 and A338539.
The asymptotic density of this sequence is 1 - 6/Pi^2 - (6/Pi^2)*A154945 - (3/Pi^2)*(A154945^2 - A324833) = 0.0033907041... (End)

			900 is in the sequence because the factorization 900 = (6*10*15) is relatively prime (since the GCD of (6,10,15) is 1) but each of the pairs (6,10), (6,15), (10,15) has a common divisor > 1. Larger examples are:
1800 = (6*15*20) = (10*12*15).
9900 = (6*10*165) = (6*15*110) = (10*15*66).
5400 = (6*20*45) = (10*12*45) = (10*15*36) = (15*18*20).
60 is not in the sequence because all its possible factorizations (4 * 15, 3 * 4 * 5, etc.) contain at least one pair that is coprime, if not more than one prime.

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

Cf. A001055, A001221, A001222, A007716, A045778, A051185, A078374, A281116, A303140, A303283, A305843, A305854, A317748, A318715, A318717, A318721.

Cf. A005117, A036785, A056170, A154945, A190641, A324833, A338539.

strfacs[n_] := If[n <= 1, {{}}, Join@@Table[(Prepend[#1, d] &)/@Select[strfacs[n/d], Min@@#1 > d &], {d, Rest[Divisors[n]]}]]; Select[Range[10000], Function[n, Select[strfacs[n], And[GCD@@# == 1, And@@(GCD[##] > 1 &)@@@Select[Tuples[#, 2], Less@@# &]] &] != {}]]
Select[Range[20000], Count[FactorInteger[#][[;;,2]], ?(#1 > 1 &)] > 2 &] (* _Amiram Eldar, Nov 01 2020 *)

cp's OEIS Frontend

A319238 Positions of zeros in A114592, the list of coefficients in the expansion of Product_{n > 1} (1 - 1/n^s).

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A319748 Number of non-isomorphic set multipartitions (multisets of sets) of weight n with empty intersection.

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A319767 Number of non-isomorphic intersecting set systems spanning n vertices whose dual is also an intersecting set system.

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A319786 Number of factorizations of n where no two factors are relatively prime.

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A066637 Total number of elements in all factorizations of n with all factors > 1.

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A301856 Number of subset-products (greater than 1) of factorizations of n into factors greater than 1.

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A303709 Number of periodic factorizations of n using elements of A007916 (numbers that are not perfect powers).

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A317748 Irregular triangle where T(n,k) is the number of factorizations of n into factors > 1 with GCD d = A027750(n, k).

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