A045778 -id:A045778 - cp's OEIS frontend

A340653 Number of balanced factorizations of n.

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

A factorization into factors > 1 is balanced if it is empty or its length is equal to its maximum Omega (A001222).

			The balanced factorizations for n = 120, 144, 192, 288, 432, 768:
  3*5*8    2*8*9    3*8*8      4*8*9      6*8*9      8*8*12
  2*2*30   3*6*8    4*6*8      6*6*8      2*8*27     2*2*8*24
  2*3*20   2*4*18   2*8*12     2*8*18     3*8*18     2*3*8*16
  2*5*12   2*6*12   4*4*12     3*8*12     4*4*27     2*4*4*24
           3*4*12   2*2*2*24   4*4*18     4*6*18     2*4*6*16
                    2*2*3*16   4*6*12     4*9*12     3*4*4*16
                               2*12*12    6*6*12     2*2*12*16
                               2*2*2*36   2*12*18    2*2*2*2*48
                               2*2*3*24   3*12*12    2*2*2*3*32
                               2*3*3*16   2*2*2*54
                                          2*2*3*36
                                          2*3*3*24
                                          3*3*3*16

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

Positions of zeros are A001358.
Positions of nonzero terms are A100959.
The co-balanced version is A340596.
Taking maximum factor instead of maximum Omega gives A340599.
The cross-balanced version is A340654.
The twice-balanced version is A340655.
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A320655 counts factorizations into semiprimes.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340597 have an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A339846, A339890, A340608.

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],#=={}||Length[#]==Max[PrimeOmega/@#]&]],{n,100}]

A340653(n, m=n, mbo=0, e=0) = if(1==n, mbo==e, sumdiv(n, d, if((d>1)&&(d<=m), A340653(n/d, d, max(mbo,bigomega(d)), 1+e)))); \\ Antti Karttunen, Oct 22 2023

Data section extended up to a(120) by Antti Karttunen, Oct 22 2023

A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

Original entry on oeis.org

0, 0, 0, 1, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 0, 4, 0, 1, 0, 1, 0, 0, 0, 3, 1, 0, 2, 1, 0, 0, 0, 6, 0, 0, 0, 4, 0, 0, 0, 3, 0, 0, 0, 1, 1, 0, 0, 7, 1, 1, 0, 1, 0, 3, 0, 3, 0, 0, 0, 2, 0, 0, 1, 10, 0, 0, 0, 1, 0, 0, 0, 10, 0, 0, 1, 1, 0, 0, 0, 7, 4, 0, 0, 2, 0, 0

Offset: 1

Gus Wiseman, Dec 27 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is counted under a(36).

			The a(1) = 0 through a(24) = 3 factorizations:
 ... 2*2 ... 2*4   3*3 .. 2*2*3 ... 2*8     . 2*3*3 . 2*2*5 ... 2*2*6
             2*2*2                  4*4                         2*3*4
                                    2*2*4                       2*2*2*3
                                    2*2*2*2

For unlabeled graphs: A140637, complement A134964.
For labeled graphs: A367867, A367868, A140638, complement A133686.
For set-systems: A367903, ranks A367907, complement A367902, ranks A367906.
For non-isomorphic set-systems: A368094, A368409, complement A368095.
For non-isomorphic multiset partitions: A368097, A355529, A368411.
Complement for non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368414.
For non-isomorphic set multipartitions: A368421, complement A368422.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412.

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[First/@FactorInteger[#]&/@#], UnsameQ@@#&]=={}&]],{n,100}]

a(n) + A368414(n) = A001055(n).

A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Original entry on oeis.org

1, 1, 0, 2, 0, 1, 0, 5, 1, 0, 0, 4, 0, 0, 0, 15, 0, 5, 0, 1, 0, 0, 0, 16, 0, 0, 8, 0, 0, 2, 0, 52, 0, 0, 0, 23, 0, 0, 0, 7, 0, 0, 0, 0, 5, 0, 0, 68, 0, 1, 0, 0, 0, 40, 0, 1, 0, 0, 0, 14, 0, 0, 1, 203, 0, 0, 0, 0, 0, 0, 0, 111, 0, 0, 4, 0, 0, 0, 0, 41, 80, 0, 0

Offset: 1

Gus Wiseman, Aug 24 2018

nonn

			The a(24) = 16 sets of sets with multiset union {1,1,2,3,4}:
  {{1},{1,2,3,4}}
  {{1,2},{1,3,4}}
  {{1,3},{1,2,4}}
  {{1,4},{1,2,3}}
  {{1},{2},{1,3,4}}
  {{1},{3},{1,2,4}}
  {{1},{4},{1,2,3}}
  {{1},{1,2},{3,4}}
  {{1},{1,3},{2,4}}
  {{1},{1,4},{2,3}}
  {{2},{1,3},{1,4}}
  {{3},{1,2},{1,4}}
  {{4},{1,2},{1,3}}
  {{1},{2},{3},{1,4}}
  {{1},{2},{4},{1,3}}
  {{1},{3},{4},{1,2}}

Andrew Howroyd, Table of n, a(n) for n = 1..500

Cf. A007716, A045778, A049311, A050326, A116540, A292432, A292444, A293511.

Cf. A318283, A318284, A318286, A318287, A318360, A318361, A318370, A318371.

nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Table[Length[sqfacs[Times@@Prime/@nrmptn[n]]],{n,90}]

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}
sig(n)={my(f=factor(n)); concat(vector(#f~, i, vector(f[i, 2], j, primepi(f[i, 1]))))}
count(sig)={my(r=0, A=O(x*x^vecmax(sig))); for(n=1, vecsum(sig)+1, my(s=0); forpart(p=n, my(q=prod(i=1, #p, 1 + x^p[i] + A)); s+=prod(i=1, #sig, polcoef(q, sig[i]))*(-1)^#p*permcount(p)); r+=(-1)^n*s/n!); r/2}
a(n)={if(n==1, 1, my(s=sig(n)); if(#s==1, s[1]==1, count(sig(n))))} \\ Andrew Howroyd, Dec 18 2018

a(n) = A050326(A181821(n)).

a(prime(n)^k) = A188445(n, k). - Andrew Howroyd, Dec 17 2018

A293511 Numbers that can be written as a product of distinct squarefree numbers in exactly one way.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 28, 29, 31, 36, 37, 41, 43, 44, 45, 47, 50, 52, 53, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 89, 92, 97, 98, 99, 100, 101, 103, 107, 109, 113, 116, 117, 120, 124, 127, 131, 137, 139, 147, 148, 149, 151, 153

Offset: 1

Gus Wiseman, Oct 11 2017

nonn

First differs from A212166 at a(128) = 363, A212166(128) = 360.

			360 is not in the sequence because it has two possible expressions: 2*3*6*10 or 2*6*30.

Cf. A001055, A005117, A045778, A050320, A050326, A292432, A292444, A293243.

nn=300;
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[nn],Length[sqfacs[#]]===1&]

A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 36, 48, 60, 72, 96, 120, 128, 144, 180, 192, 216, 240, 256, 288, 360, 384, 420, 432, 480, 576, 720, 768, 840, 864, 900, 960, 1024, 1080, 1152, 1260, 1440, 1680, 1728, 1800, 1920, 2048, 2160, 2304, 2520, 2592, 2880, 3072, 3360, 3456, 3600

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

This is the sorted list of positions of first appearances in A001055 of each element of the range (A045782).

			Factorizations of n for n = 4, 8, 12, 16, 24, 36, 48, 60:
  4    8      12     16       24       36       48         60
  2*2  2*4    2*6    2*8      3*8      4*9      6*8        2*30
       2*2*2  3*4    4*4      4*6      6*6      2*24       3*20
              2*2*3  2*2*4    2*12     2*18     3*16       4*15
                     2*2*2*2  2*2*6    3*12     4*12       5*12
                              2*3*4    2*2*9    2*3*8      6*10
                              2*2*2*3  2*3*6    2*4*6      2*5*6
                                       3*3*4    3*4*4      3*4*5
                                       2*2*3*3  2*2*12     2*2*15
                                                2*2*2*6    2*3*10
                                                2*2*3*4    2*2*3*5
                                                2*2*2*2*3

R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

All terms belong to A025487
Includes all highly factorable numbers A033833.
Factorizations are A001055, with image A045782.
The least number with A045782(n) factorizations is A045783(n).
The least number with n factorizations is A330973(n).
The strict version is A330997.
Cf. A001222, A002033, A007716, A045778, A318284, A325238, A330935, A330936, A330976, A330977, A330989, A330991, A330992.

nn=1000;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[facs,nn];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

A033833 Highly factorable numbers: numbers with a record number of proper factorizations.

Original entry on oeis.org

1, 4, 8, 12, 16, 24, 36, 48, 72, 96, 120, 144, 192, 216, 240, 288, 360, 432, 480, 576, 720, 960, 1080, 1152, 1440, 2160, 2880, 4320, 5040, 5760, 7200, 8640, 10080, 11520, 12960, 14400, 15120, 17280, 20160, 25920, 28800, 30240, 34560

Offset: 1

Jeff Burch

nonn

First differs from A045783 and A330972 in lacking 60.

Indices of records in A028422 or A001055.

			From _Gus Wiseman_, Jan 13 2020: (Start)
Factorizations of the initial terms:
  ()  (4)    (8)      (12)     (16)       (24)       (36)       (48)
      (2*2)  (2*4)    (2*6)    (2*8)      (3*8)      (4*9)      (6*8)
             (2*2*2)  (3*4)    (4*4)      (4*6)      (6*6)      (2*24)
                      (2*2*3)  (2*2*4)    (2*12)     (2*18)     (3*16)
                               (2*2*2*2)  (2*2*6)    (3*12)     (4*12)
                                          (2*3*4)    (2*2*9)    (2*3*8)
                                          (2*2*2*3)  (2*3*6)    (2*4*6)
                                                     (3*3*4)    (3*4*4)
                                                     (2*2*3*3)  (2*2*12)
                                                                (2*2*2*6)
                                                                (2*2*3*4)
                                                                (2*2*2*2*3)
(End)

Amiram Eldar, Table of n, a(n) for n = 1..235 (terms 1..118 from E. R. Canfield et al.)
E. R. Canfield, P. Erdős, C. Pomerance, On a problem of Oppenheim concerning Factorisatio Numerorum, J. Number Theory 17 (1983) 1-28, Table 1, column "n".
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.
Arnold Knopfmacher and Michael Mays, Ordered and unordered factorizations of integers, Mathematica Journal, Vol. 10, No. 1 (2006), pp. 72-89.

All terms belong to A025487 as well as to A330972.
The corresponding records are A272691.
The strict version is A331200.
Factorizations are A001055, with image A045782 and complement A330976.
Cf. A028422, A033834, A045778, A045783, A330973, A330998, A331049, A331050.

nn=100;
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[facs[n]],{n,nn}];
Table[Position[qv,i][[1,1]],{i,qv//.{foe___,x_,y_,afe___}/;x>=y:>{foe,x,afe}}] (* Gus Wiseman, Jan 13 2020 *)

A001055(a(n)) = A272691(n). - Gus Wiseman, Jan 13 2020

A293243 Numbers that cannot be written as a product of distinct squarefree numbers.

Original entry on oeis.org

4, 8, 9, 16, 24, 25, 27, 32, 40, 48, 49, 54, 56, 64, 72, 80, 81, 88, 96, 104, 108, 112, 121, 125, 128, 135, 136, 144, 152, 160, 162, 169, 176, 184, 189, 192, 200, 208, 216, 224, 232, 240, 243, 248, 250, 256, 272, 288, 289, 296, 297, 304, 320, 324, 328, 336

Offset: 1

Gus Wiseman, Oct 03 2017

nonn

First differs from A212164 at a(441).
Numbers n such that A050326(n) = 0. - Felix Fröhlich, Oct 04 2017
Includes A246547, and all numbers of the form p^a*q^b where p and q are primes, a >= 1 and b >= 3. - Robert Israel, Oct 10 2017
Also numbers whose prime indices cannot be partitioned into a set of sets. For example, the prime indices of 90 are {1,2,2,3}, and we have sets of sets: {{2},{1,2,3}}, {{1,2},{2,3}}, {{1},{2},{2,3}}, {{2},{3},{1,2}}, so 90 is not in the sequence. - Gus Wiseman, Apr 28 2025

			120 is not in the sequence because 120 = 2*6*10. 3600 is not in the sequence because 3600 = 2*6*10*30.

Robert Israel, Table of n, a(n) for n = 1..10000

These are the zeros of A050326.
Multiset partitions of this type (set of sets) are counted by A050342.
Twice-partitions of this type (set of sets) are counted by A279785, see also A358914.
Normal multisets of this type are counted by A292432, A292444, A381996, A382214.
The case of a unique choice is A293511, counted by A382079.
For distinct block-sums instead of blocks see A381806, A381990, A381992, A382075.
Partitions of this type are counted by A382078.
The complement is A382200, counted by A382077.
A001055 counts factorizations, strict A045778.
A050320 counts factorizations into squarefree numbers.
A050345 counts factorizations partitioned into into distinct sets.
A317141 counts coarsenings of prime indices, refinements A300383.
Cf. A000720, A001222, A005117, A089259, A116539, A246547, A270995, A381441, A382201, A382216.

N:= 1000: # to get all terms <= N
A:= Vector(N):
A[1]:= 1:
for n from 2 to N do
  if numtheory:-issqrfree(n) then
      S:= [$1..N/n]; T:= n*S; A[T]:= A[T]+A[S]
    fi;
od:
select(t -> A[t]=0, [$1..N]); # Robert Israel, Oct 10 2017

nn=500;
sqfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[sqfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],SquareFreeQ]}]];
Select[Range[nn],Length[sqfacs[#]]===0&]

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each 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, 1, 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, 6, 1, 2, 3, 3, 2, 5, 1, 5, 1, 2, 1, 9, 2, 2, 2

Offset: 1

Gus Wiseman, Dec 29 2023

nonn

For example, the factorization f = 2*3*6 has two ways to choose a prime factor of each factor, namely (2,3,2) and (2,3,3), but neither of these has all different elements, so f is not counted under a(36).

			The a(n) factorizations for selected n:
  1    6      12     24      30       60        72      120
       2*3    2*6    2*12    2*15     2*30      2*36    2*60
              3*4    3*8     3*10     3*20      3*24    3*40
                     4*6     5*6      4*15      4*18    4*30
                             2*3*5    5*12      6*12    5*24
                                      6*10      8*9     6*20
                                      2*3*10            8*15
                                      2*5*6             10*12
                                      3*4*5             2*3*20
                                                        2*5*12
                                                        2*6*10
                                                        3*4*10
                                                        3*5*8
                                                        4*5*6

For labeled graphs: A133686, complement A367867, A367868, A140638.
For unlabeled graphs: A134964, complement A140637.
For set-systems: A367902, ranks A367906, complement A367903, ranks A367907.
For non-isomorphic set-systems: A368095, complement A368094, A368409.
Complementary non-isomorphic multiset partitions: A368097, A355529, A368411.
For non-isomorphic multiset partitions: A368098, A368100.
The complement is counted by A368413.
For non-isomorphic set multipartitions: A368422, complement A368421.
For divisors instead of prime factors: A370813, complement A370814.
A001055 counts factorizations, strict A045778.
A007716 counts non-isomorphic multiset partitions, connected A007718.
A058891 counts set-systems, unlabeled A000612, connected A323818.
A283877 counts non-isomorphic set-systems, connected A300913.
Cf. A340596, A340653, A367769, A367901, A368187, A368412, A368413.

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[First/@FactorInteger[#]&/@#], UnsameQ@@#&]!={}&]],{n,100}]

a(n) = A001055(n) - A368413(n).

A379721 Numbers whose prime indices have sum <= product.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 11, 13, 15, 17, 19, 21, 23, 25, 27, 29, 30, 31, 33, 35, 37, 39, 41, 42, 43, 45, 47, 49, 50, 51, 53, 54, 55, 57, 59, 61, 63, 65, 66, 67, 69, 70, 71, 73, 75, 77, 78, 79, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95, 97, 98, 99, 100, 101, 102, 103

Offset: 1

Gus Wiseman, Jan 05 2025

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.
Partitions of this type are counted by A319005.
The complement is A325038.

			The terms together with their prime indices begin:
    1: {}
    2: {1}
    3: {2}
    5: {3}
    7: {4}
    9: {2,2}
   11: {5}
   13: {6}
   15: {2,3}
   17: {7}
   19: {8}
   21: {2,4}
   23: {9}
   25: {3,3}
   27: {2,2,2}
   29: {10}
   30: {1,2,3}

The case of equality is A301987, inequality A325037.
Nonpositive positions in A325036.
A000040 lists the primes, differences A001223.
A055396 gives least prime index, greatest A061395.
A056239 adds up prime indices, row sums of A112798, counted by A001222.
A379681 gives sum plus product of prime indices, firsts A379682.
Counting and ranking multisets by comparing sum and product:
- same: A001055 (strict A045778), ranks A301987
- divisible: A057567, ranks A326155
- divisor: A057568, ranks A326149, see A326156, A326172, A379733
- greater: A096276 shifted right, ranks A325038
- greater or equal: A096276, ranks A325044
- less: A114324, ranks A325037, see A318029
- less or equal: A319005, ranks A379721 (this)
- different: A379736, ranks A379722, see A111133
Cf. A000720, A003963, A046022, A075254, A075255, A178503, A175508, A319000, A325034, A325035, A379720.

prix[n_]:=If[n==1,{},Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]];
Select[Range[100],Total[prix[#]]<=Times@@prix[#]&]

Number k such that A056239(k) <= A003963(k).

A320892 Numbers with an even number of prime factors (counted with multiplicity) that cannot be factored into distinct semiprimes.

Original entry on oeis.org

16, 64, 81, 96, 144, 160, 224, 256, 324, 352, 384, 400, 416, 486, 544, 576, 608, 625, 640, 729, 736, 784, 864, 896, 928, 960, 992, 1024, 1184, 1215, 1296, 1312, 1344, 1376, 1408, 1440, 1504, 1536, 1600, 1664, 1696, 1701, 1888, 1936, 1944, 1952, 2016, 2025

Offset: 1

Gus Wiseman, Oct 23 2018

nonn

A semiprime (A001358) is a product of any two not necessarily distinct primes.
If A025487(k) is in the sequence then so is every number with the same prime signature. - David A. Corneth, Oct 23 2018
Numbers for which A001222(n) is even and A322353(n) is zero. - Antti Karttunen, Dec 06 2018

			A complete list of all factorizations of 1296 into semiprimes is:
  1296 = (4*4*9*9)
  1296 = (4*6*6*9)
  1296 = (6*6*6*6)
None of these is strict, so 1296 belongs to the sequence.

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A001055, A001358, A005117, A006881, A007717, A025487, A028260, A045778, A318871, A318953, A320462, A320655, A320656, A320891, A320893, A320894, A322353.

strsemfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strsemfacs[n/d],Min@@#>d&]],{d,Select[Rest[Divisors[n]],PrimeOmega[#]==2&]}]];
Select[Range[1000],And[EvenQ[PrimeOmega[#]],strsemfacs[#]=={}]&]

A322353(n, m=n, facs=List([])) = if(1==n, my(u=apply(bigomega,Vec(facs))); (0==length(u)||(2==vecmin(u)&&2==vecmax(u))), my(s=0, newfacs); fordiv(n, d, if((d>1)&&(d<=m), newfacs = List(facs); listput(newfacs,d); s += A322353(n/d, d-1, newfacs))); (s));
isA300892(n) = if(bigomega(n)%2,0,(0==A322353(n))); \\ Antti Karttunen, Dec 06 2018

cp's OEIS Frontend

A340653 Number of balanced factorizations of n.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

A368413 Number of factorizations of n into positive integers > 1 such that it is not possible to choose a different prime factor of each factor.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A318361 Number of strict set multipartitions (sets of sets) of a multiset whose multiplicities are the prime indices of n.

Views

Author

Keywords

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A293511 Numbers that can be written as a product of distinct squarefree numbers in exactly one way.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A330972 Sorted list containing the least number with each possible nonzero number of factorizations into factors > 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

A033833 Highly factorable numbers: numbers with a record number of proper factorizations.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula

A293243 Numbers that cannot be written as a product of distinct squarefree numbers.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

A368414 Number of factorizations of n into positive integers > 1 such that it is possible to choose a different prime factor of each factor.

Views

Author

Keywords

Comments