A330976 -id:A330976 - cp's OEIS frontend

A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

1, 4, 12, 0, 72, 0, 480

Offset: 0

Gus Wiseman, Jan 07 2020

			The A001055(n) factorizations for n = 1, 4, 12, 72:
  ()  (4)    (12)     (72)
      (2*2)  (2*6)    (8*9)
             (3*4)    (2*36)
             (2*2*3)  (3*24)
                      (4*18)
                      (6*12)
                      (2*4*9)
                      (2*6*6)
                      (3*3*8)
                      (3*4*6)
                      (2*2*18)
                      (2*3*12)
                      (2*2*2*9)
                      (2*2*3*6)
                      (2*3*3*4)
                      (2*2*2*3*3)

All nonzero terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782.
The least number with exactly n factorizations is A330973(n).
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly prime(n) factorizations is A330992(n).
Cf. A002033, A045778, A045783, A318284, A330935, A330972, A330976, A330990, A330991, A331022.

A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

Original entry on oeis.org

1, 1, 2, 2, 3, 4, 5, 7, 5, 7, 9, 12, 11, 11, 16, 19, 21, 15, 29, 26, 30, 15, 31, 38, 22, 47, 52, 45, 36, 57, 64, 30, 77, 98, 67, 74, 97, 66, 105, 42, 109, 118, 92, 109, 171, 97, 141, 162, 137, 165, 56, 212, 181, 52, 198, 189, 289, 139, 250, 257, 269, 254, 77, 382, 267

Offset: 1

Christian G. Bower, Oct 15 1999

nonn

For A025487(m) = 2^k = A000079(k), we have a(m) = A000041(k).

Is a(k) = A000110(k) for A025487(m) = A002110(k)?

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(11) = 9 factorizations:
  {}  2  4    6    8      12     16       24       30     32         36
         2*2  2*3  2*4    2*6    2*8      3*8      5*6    4*8        4*9
                   2*2*2  3*4    4*4      4*6      2*15   2*16       6*6
                          2*2*3  2*2*4    2*12     3*10   2*2*8      2*18
                                 2*2*2*2  2*2*6    2*3*5  2*4*4      3*12
                                          2*3*4           2*2*2*4    2*2*9
                                          2*2*2*3         2*2*2*2*2  2*3*6
                                                                     3*3*4
                                                                     2*2*3*3
(End)

R. J. Mathar and Michael De Vlieger, Table of n, a(n) for n = 1..5000 (First 300 terms from _R. J. Mathar_)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

Cf. A000041, A000079, A000110, A001055, A002110, A025487.
The version indexed by unsorted prime signature is A331049.
The version indexed by prime shadow (A181819, A181821) is A318284.
This sequence has range A045782 (same as A001055).
Cf. A033833, A045778, A045783, A070175, A181821, A325238, A330972, A330973, A330976, A330989, A330990, A330998, A331050.

A050322 := proc(n)
    A001055(A025487(n)) ;
end proc: # R. J. Mathar, May 25 2017

c[1, r_] := c[1, r] = 1; c[n_, r_] := c[n, r] = Module[{d, i}, d = Select[Divisors[n], 1 < # <= r &]; Sum[c[n/d[[i]], d[[i]]], {i, 1, Length[d]}]]; Map[c[#, #] &, Union@ Table[Times @@ MapIndexed[If[n == 1, 1, Prime[First@ #2]]^#1 &, Sort[FactorInteger[n][[All, -1]], Greater]], {n, Product[Prime@ i, {i, 6}]}]] (* Michael De Vlieger, Jul 10 2017, after Dean Hickerson at A001055 *)
facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Length/@facs/@First/@GatherBy[Range[1000],If[#==1,{},Sort[Last/@FactorInteger[#]]]&] (* Gus Wiseman, Jan 13 2020 *)

A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

Original entry on oeis.org

1, 3, 5, 6, 7, 9, 10, 12, 14, 15, 16, 18, 19, 20, 21, 22, 23, 24, 25, 27, 28, 29, 30, 31, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 47, 49, 51, 52, 53, 54, 56, 57, 58, 59, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 77, 78, 79

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

This is the sorted list of positions of first appearances in A318284 of each element of the range A045782.

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. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			Factorizations of the inverse prime shadows of the initial terms:
    4    8      12     16       36       24       60       48
    2*2  2*4    2*6    2*8      4*9      3*8      2*30     6*8
         2*2*2  3*4    4*4      6*6      4*6      3*20     2*24
                2*2*3  2*2*4    2*18     2*12     4*15     3*16
                       2*2*2*2  3*12     2*2*6    5*12     4*12
                                2*2*9    2*3*4    6*10     2*3*8
                                2*3*6    2*2*2*3  2*5*6    2*4*6
                                3*3*4             3*4*5    3*4*4
                                2*2*3*3           2*2*15   2*2*12
                                                  2*3*10   2*2*2*6
                                                  2*2*3*5  2*2*3*4
                                                           2*2*2*2*3
The corresponding multiset partitions:
    {11}    {111}      {112}      {1111}        {1122}        {1112}
    {1}{1}  {1}{11}    {1}{12}    {1}{111}      {1}{122}      {1}{112}
            {1}{1}{1}  {2}{11}    {11}{11}      {11}{22}      {11}{12}
                       {1}{1}{2}  {1}{1}{11}    {12}{12}      {2}{111}
                                  {1}{1}{1}{1}  {2}{112}      {1}{1}{12}
                                                {1}{1}{22}    {1}{2}{11}
                                                {1}{2}{12}    {1}{1}{1}{2}
                                                {2}{2}{11}
                                                {1}{1}{2}{2}

Taking n instead of the inverse prime shadow of n gives A330972.
Factorizations are A001055, with image A045782, with complement A330976.
Factorizations of inverse prime shadows are A318284.
Cf. A025487, A033833, A045778, A045783, A181819, A181821, A318286, A325238, A330973, A330989, A330990, A330993, A330997.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
nds=Table[Length[facs[Times@@Prime/@nrmptn[n]]],{n,50}];
Table[Position[nds,i][[1,1]],{i,First/@Gather[nds]}]

A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

Original entry on oeis.org

4, 8, 16, 24, 60, 0, 0, 96, 0, 144, 216, 0, 0, 0, 288, 0, 0, 0, 768, 0, 0, 0, 0, 0, 864, 8192, 0, 0, 1080, 0, 0, 0, 1800, 3072, 0, 0, 0, 0, 0, 0, 0, 2304, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3456, 0, 3600, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 24576

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

			Factorizations of the initial positive terms are:
  4    8      16       24       60       96
  2*2  2*4    2*8      3*8      2*30     2*48
       2*2*2  4*4      4*6      3*20     3*32
              2*2*4    2*12     4*15     4*24
              2*2*2*2  2*2*6    5*12     6*16
                       2*3*4    6*10     8*12
                       2*2*2*3  2*5*6    2*6*8
                                3*4*5    3*4*8
                                2*2*15   4*4*6
                                2*3*10   2*2*24
                                2*2*3*5  2*3*16
                                         2*4*12
                                         2*2*3*8
                                         2*2*4*6
                                         2*3*4*4
                                         2*2*2*12
                                         2*2*2*2*6
                                         2*2*2*3*4
                                         2*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 positive terms belong to A025487 and also A033833.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of partitions is prime are A046063.
Numbers whose number of strict partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers with a prime number of factorizations are A330991.
The least number with exactly 2^n factorizations is A330989(n).
Cf. A001222, A045783, A325238, A330972, A330973, A330976, A330993, A330998.

More terms from Jinyuan Wang, Jul 07 2021

A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

Original entry on oeis.org

1, 2, 3, 4, 6, 15, 44

Offset: 1

Gus Wiseman, Jan 07 2020

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. The inverse prime shadow of n is the least number whose prime exponents are the prime indices of n.

			The factorizations of A181821(n) for n = 1, 2, 3, 4, 6, 15:
  ()  (2)  (4)    (6)    (12)     (72)
           (2*2)  (2*3)  (2*6)    (8*9)
                         (3*4)    (2*36)
                         (2*2*3)  (3*24)
                                  (4*18)
                                  (6*12)
                                  (2*4*9)
                                  (2*6*6)
                                  (3*3*8)
                                  (3*4*6)
                                  (2*2*18)
                                  (2*3*12)
                                  (2*2*2*9)
                                  (2*2*3*6)
                                  (2*3*3*4)
                                  (2*2*2*3*3)

The same for prime numbers (instead of powers of 2) is A330993,
Factorizations are A001055, with image A045782.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with exactly 2^n factorizations is A330989.
Cf. A033833, A045778, A045783, A181821, A305936, A318283, A318284, A330972, A330973, A330976, A330998, A331022.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nrmptn[n_]:=Join@@MapIndexed[Table[#2[[1]],{#1}]&,If[n==1,{},Flatten[Cases[FactorInteger[n]//Reverse,{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[100],IntegerQ[Log[2,Length[facs[Times@@Prime/@nrmptn[#]]]]]&]

A001055(A181821(a(n))) = 2^k for some k >= 0.

A331200 Least number with each record number of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 6, 12, 24, 48, 60, 96, 120, 180, 240, 360, 480, 720, 840, 1080, 1260, 1440, 1680, 2160, 2520, 3360, 4320, 5040, 7560, 8640, 10080, 15120, 20160, 25200, 30240, 40320, 45360, 50400, 55440, 60480, 75600, 90720, 100800, 110880, 120960, 151200, 181440, 221760

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A330997 in lacking 64.

			Strict factorizations of the initial terms:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

A subset of A330997.
All terms belong to A025487.
This is the strict version of highly factorable numbers A033833.
The corresponding records are A331232(n) = A045778(a(n)).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n)
Cf. A045783, A325238, A330972, A330973, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Table[Position[qv,i][[1,1]],{i,Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]}]

a(37) and beyond from Giovanni Resta, Jan 17 2020

A331232 Record numbers of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 2, 3, 5, 7, 9, 10, 16, 18, 25, 34, 38, 57, 59, 67, 70, 91, 100, 117, 141, 161, 193, 253, 296, 306, 426, 552, 685, 692, 960, 1060, 1067, 1216, 1220, 1589, 1591, 1912, 2029, 2157, 2524, 2886, 3249, 3616, 3875, 4953, 5147, 5285, 5810, 6023, 6112, 6623, 8129

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

			Representatives for the initial records and their strict factorizations:
  ()  (6)    (12)   (24)     (48)     (60)      (96)      (120)
      (2*3)  (2*6)  (3*8)    (6*8)    (2*30)    (2*48)    (2*60)
             (3*4)  (4*6)    (2*24)   (3*20)    (3*32)    (3*40)
                    (2*12)   (3*16)   (4*15)    (4*24)    (4*30)
                    (2*3*4)  (4*12)   (5*12)    (6*16)    (5*24)
                             (2*3*8)  (6*10)    (8*12)    (6*20)
                             (2*4*6)  (2*5*6)   (2*6*8)   (8*15)
                                      (3*4*5)   (3*4*8)   (10*12)
                                      (2*3*10)  (2*3*16)  (3*5*8)
                                                (2*4*12)  (4*5*6)
                                                          (2*3*20)
                                                          (2*4*15)
                                                          (2*5*12)
                                                          (2*6*10)
                                                          (3*4*10)
                                                          (2*3*4*5)

Giovanni Resta, Table of n, a(n) for n = 1..122
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

The non-strict version is A272691.
The first appearance of a(n) in A045778 is at index A331200(n).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with n strict factorizations is A330974(n).
The least number with A045779(n) strict factorizations is A045780(n).
Cf. A033833, A045783, A325238, A330972, A330973, A330997, A331023/A331024, A331201.

nn=1000;
strfacs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[strfacs[n/d],Min@@#>d&]],{d,Rest[Divisors[n]]}]];
qv=Table[Length[strfacs[n]],{n,nn}];
Union[qv//.{foe___,x_,y_,afe___}/;x>y:>{foe,x,afe}]

def fact(num):
    ret = []
    temp = num
    div = 2
    while temp > 1:
        while temp % div == 0:
            ret.append(div)
            temp //= div
        div += 1
    return ret
def all_partitions(lst):
    if lst:
        x = lst[0]
        for partition in all_partitions(lst[1:]):
            yield [x] + partition
            for i, _ in enumerate(partition):
                partition[i] *= x
                yield partition
                partition[i] //= x
    else:
        yield []
best = 0
terms = [0]
q = 2
while len(terms) < 100:
    total_set = set()
    factors = fact(q)
    total_set = set(tuple(sorted(x)) for x in all_partitions(factors) if len(x) == len(set(x)))
    if len(total_set) > best:
        best = len(total_set)
        terms.append(best)
        print(q,best)
    q += 2#only check evens
print(terms)
#  David Consiglio, Jr., Jan 14 2020

a(n) = A045778(A331200(n)).

a(26)-a(37) from David Consiglio, Jr., Jan 14 2020

a(38) and beyond from Giovanni Resta, Jan 17 2020

A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

Original entry on oeis.org

3, 4, 5, 7, 8, 10, 11, 12, 13, 21, 22, 25, 33, 38, 41, 45, 46, 49, 50, 55, 57, 58, 63

Offset: 1

Gus Wiseman, Jan 07 2020

This multiset (row k of A305936) is generally not the same as the multiset of prime indices of k. For example, the prime indices of 12 are {1,1,2}, while a multiset whose multiplicities are {1,1,2} is {1,1,2,3}.

Also numbers whose inverse prime shadow has a prime number of factorizations. A prime index of k is a number m such that prime(m) divides k. The multiset of prime indices of k is row k of A112798. The inverse prime shadow of k is the least number whose prime exponents are the prime indices of k.

			The multiset partitions for n = 1..6:
  {11}    {12}    {111}      {1111}        {123}      {1112}
  {1}{1}  {1}{2}  {1}{11}    {1}{111}      {1}{23}    {1}{112}
                  {1}{1}{1}  {11}{11}      {2}{13}    {11}{12}
                             {1}{1}{11}    {3}{12}    {2}{111}
                             {1}{1}{1}{1}  {1}{2}{3}  {1}{1}{12}
                                                      {1}{2}{11}
                                                      {1}{1}{1}{2}
The factorizations for n = 1..8:
  4    6    8      16       30     24       32         60
  2*2  2*3  2*4    2*8      5*6    3*8      4*8        2*30
            2*2*2  4*4      2*15   4*6      2*16       3*20
                   2*2*4    3*10   2*12     2*2*8      4*15
                   2*2*2*2  2*3*5  2*2*6    2*4*4      5*12
                                   2*3*4    2*2*2*4    6*10
                                   2*2*2*3  2*2*2*2*2  2*5*6
                                                       3*4*5
                                                       2*2*15
                                                       2*3*10
                                                       2*2*3*5

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

The same for powers of 2 (instead of primes) is A330990.
Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of integer partitions is prime are A046063.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of set partitions is prime are A051130.
Numbers whose number of factorizations is a power of 2 are A330977.
The least number with prime(n) factorizations is A330992(n).
Factorizations of a number's inverse prime shadow are A318284.
Numbers with a prime number of factorizations are A330991.
Cf. A033833, A045783, A056239, A181819, A181821, A305936, A318286, A325755, A330972, A330973, A330998.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
unsh[n_]:=Times@@MapIndexed[Prime[#2[[1]]]^#1&,Reverse[Flatten[Cases[FactorInteger[n],{p_,k_}:>Table[PrimePi[p],{k}]]]]];
Select[Range[30],PrimeQ[Length[facs[unsh[#]]]]&]

A001055(A181821(a(n))) belongs to A000040.

A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.

Original entry on oeis.org

6, 8, 10, 12, 14, 15, 16, 18, 20, 21, 22, 24, 26, 27, 28, 30, 32, 33, 34, 35, 36, 38, 39, 40, 42, 44, 45, 46, 48, 50, 51, 52, 54, 55, 56, 57, 58, 62, 63, 65, 66, 68, 69, 70, 74, 75, 76, 77, 78, 80, 81, 82, 85, 86, 87, 88, 91, 92, 93, 94, 95, 98, 99, 100, 102

Offset: 1

Gus Wiseman, Jan 12 2020

nonn

First differs from A080257 in lacking 60.

			Strict factorizations of selected terms:
  (6)    (12)   (24)     (48)     (216)
  (2*3)  (2*6)  (3*8)    (6*8)    (3*72)
         (3*4)  (4*6)    (2*24)   (4*54)
                (2*12)   (3*16)   (6*36)
                (2*3*4)  (4*12)   (8*27)
                         (2*3*8)  (9*24)
                         (2*4*6)  (12*18)
                                  (2*108)
                                  (3*8*9)
                                  (4*6*9)
                                  (2*3*36)
                                  (2*4*27)
                                  (2*6*18)
                                  (2*9*12)
                                  (3*4*18)
                                  (3*6*12)
                                  (2*3*4*9)

The version for strict integer partitions is A035359.
The version for integer partitions is A046063.
The version for set partitions is A051130.
The non-strict version is A330991.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Numbers whose number of strict factorizations is odd are A331230.
Numbers whose number of strict factorizations is even are A331231.
The least number with n strict factorizations is A330974(n).
Cf. A001318, A045780, A318286, A328966, A330992, A330993, A330997, A331023/A331024, A331050, A331051, A331200, A331232.

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

A272691 Number of factorizations of the highly factorable numbers A033833.

Original entry on oeis.org

1, 2, 3, 4, 5, 7, 9, 12, 16, 19, 21, 29, 30, 31, 38, 47, 52, 57, 64, 77, 98, 105, 109, 118, 171, 212, 289, 382, 392, 467, 484, 662, 719, 737, 783, 843, 907, 1097, 1261, 1386, 1397, 1713, 1768, 2116, 2179, 2343, 3079, 3444, 3681, 3930, 5288, 5413, 5447

Offset: 1

N. J. A. Sloane, Jun 02 2016, following a suggestion from George Beck

nonn

These are defined as record numbers of factorizations (A001055). - Gus Wiseman, Jan 13 2020

			From _Gus Wiseman_, Jan 13 2020: (Start)
The a(1) = 1 through a(8) = 12 factorizations of highly factorable numbers:
  ()  (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.)
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.
Jun Kyo Kim, On highly factorable numbers, Journal Of Number Theory, Vol. 72, No. 1 (1998), pp. 76-91.

The strict version is A331232.
Factorizations are A001055, with image A045782 and complement A330976.
Highly factorable numbers are A033833, with strict version A331200.
Cf. A033834, A045778, A045783, A330972, A330973, A330998.

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

a(n) = A001055(A033833(n)).

a(n) = A033834(n) + 1. - Amiram Eldar, Jun 07 2019

cp's OEIS Frontend

A330989 Least positive integer with exactly 2^n factorizations into factors > 1, or 0 if no such integer exists.

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A050322 Number of factorizations indexed by prime signatures: A001055(A025487).

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A330998 Sorted list containing the least number whose inverse prime shadow (A181821) has each possible nonzero number of factorizations into factors > 1.

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A330992 Least positive integer with exactly prime(n) factorizations into factors > 1, or 0 if no such integer exists.

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A330990 Numbers whose inverse prime shadow (A181821) has its number of factorizations into factors > 1 (A001055) equal to a power of 2 (A000079).

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A331200 Least number with each record number of factorizations into distinct factors > 1.

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A331232 Record numbers of factorizations into distinct factors > 1.

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A330993 Numbers k such that a multiset whose multiplicities are the prime indices of k has a prime number of multiset partitions.

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A331201 Numbers k such that the number of factorizations of k into distinct factors > 1 is a prime number.