A033833 -id:A033833 - cp's OEIS frontend

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 24, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 40, 42, 46, 49, 51, 54, 55, 56, 57, 58, 60, 62, 64, 65, 66, 69, 70, 74, 77, 78, 81, 82, 84, 85, 86, 87, 88, 90, 91, 93, 94, 95, 96, 102, 104, 105, 106, 110, 111, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

In short, A001055(a(n)) belongs to A000040.

			Factorizations of selected terms:
  (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)

Amiram Eldar, Table of n, a(n) for n = 1..10000
R. E. Canfield, P. Erdős and C. Pomerance, On a Problem of Oppenheim concerning "Factorisatio Numerorum", J. Number Theory 17 (1983), 1-28.

Factorizations are A001055, with image A045782, with complement A330976.
Numbers whose number of strict integer partitions is prime are A035359.
Numbers whose number of integer partitions is prime are A046063.
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).
Cf. A001222, A033833, A045783, A181819, A181821, A326622, A330972, A330973, A330993.

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

A045780 Least value with A045779(n) factorizations into distinct factors.

Original entry on oeis.org

1, 6, 12, 64, 24, 256, 48, 512, 60, 96, 2048, 144, 210, 120, 216, 180, 384, 288, 16384, 240, 432, 420, 65536, 1536, 360, 480, 900, 864, 3072, 1152, 1296, 2310, 524288, 6144, 960, 720, 840, 2304, 1728, 1080, 1260, 2592, 2097152, 1800, 4608, 24576, 4194304, 1440, 3456

Offset: 1

David W. Wilson

nonn

			From _Gus Wiseman_, Jan 11 2020: (Start)
The strict factorizations of a(n) for n = 1..9:
  ()  (6)    (12)   (64)     (24)     (256)     (48)     (512)     (60)
      (2*3)  (2*6)  (2*32)   (3*8)    (4*64)    (6*8)    (8*64)    (2*30)
             (3*4)  (4*16)   (4*6)    (8*32)    (2*24)   (16*32)   (3*20)
                    (2*4*8)  (2*12)   (2*128)   (3*16)   (2*256)   (4*15)
                             (2*3*4)  (2*4*32)  (4*12)   (4*128)   (5*12)
                                      (2*8*16)  (2*3*8)  (2*4*64)  (6*10)
                                                (2*4*6)  (2*8*32)  (2*5*6)
                                                         (4*8*16)  (3*4*5)
                                                                   (2*3*10)
(End)
30 is not in the sequence even though A045779(30) = 5. As 24 is the smallest k such that A045779(k) = 5 we have a(m) = 24 where m is such that A045779(m) = 5 which turns out to be m = 5 (not every positive integer is in A045779). So a(5) = 24. - _David A. Corneth_, Oct 24 2024

David A. Corneth, Table of n, a(n) for n = 1..953

All terms belong to A025487.
The non-strict version is A045783.
The sorted version is A330997.
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with exactly n strict factorizations is A330974(n).
Cf. A033833, A318286, A330972, A330973, A330989.

More terms from David A. Corneth, Oct 24 2024

A330974 Least positive integer with n factorizations into distinct factors > 1, and 0 if no such number exists.

Original entry on oeis.org

1, 6, 12, 64, 24, 256, 48, 512, 60, 96, 0, 2048, 0, 144, 210, 120, 216, 180, 384, 0, 288, 16384, 0, 0, 240, 0, 432, 0, 0, 0, 420, 65536, 1536, 360, 0, 0, 0, 480, 0, 900, 0, 864, 3072, 1152, 0, 1296, 0, 0, 0, 0, 0, 2310, 0, 524288, 6144, 960, 720, 0, 840, 0, 2304

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

All nonzero terms belong to A025487.
Strict factorizations are A045778, with image A045779.
The version with zeros removed is A045780.
The non-strict version is A330973.
Positions of zeros are A330975.
The sorted version is A330997.
Cf. A001055, A001222, A033833, A045782, A045783, A318286, A328966, A330972.

nn=10;
fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[Select[fam[#],UnsameQ@@#&]&,2^nn];
Table[If[#=={},0,#[[1,1]]]&[Position[nds,i]],{i,nn}]

More terms from Jinyuan Wang, Jul 07 2021

A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 9, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, 51, 52, 53, 55, 57, 58, 59, 61, 62, 63, 65, 67, 68, 69, 71, 72, 73, 74, 75, 76, 77, 79, 82, 83, 85, 86, 87

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

The complement starts: 8, 16, 24, 27, 30, 32, 36, 40.

			Factorizations of 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)

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 strict integer partitions is A331022.
Factorizations are A001055, with image A045782.
The least number with exactly n factorizations is A330973(n).
The least number with exactly 2^n factorizations is A330989(n).
Numbers whose inverse prime shadow belongs to this sequence are A330990.
Numbers with a prime number of factorizations are A330991.
Cf. A002033, A033833, A045778, A045783, A317145, A326028, A330972, A330976.

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

A330997 Sorted list containing the least number with each possible nonzero number of factorizations into distinct factors > 1.

Original entry on oeis.org

1, 6, 12, 24, 48, 60, 64, 96, 120, 144, 180, 210, 216, 240, 256, 288, 360, 384, 420, 432, 480, 512, 720, 840, 864, 900, 960, 1080, 1152, 1260, 1296, 1440, 1536, 1680, 1728, 1800, 2048, 2160, 2304, 2310, 2520, 2592, 2880, 3072, 3360, 3456, 3600, 3840, 4320

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

			The strict factorizations of a(n) for n = 1..9.
  {}  6    12   24     48     60      64     96      120
      2*3  2*6  3*8    6*8    2*30    2*32   2*48    2*60
           3*4  4*6    2*24   3*20    4*16   3*32    3*40
                2*12   3*16   4*15    2*4*8  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

All terms belong to A025487.
Strict factorizations are A045778, with image A045779.
The unsorted version is A045780.
The non-strict version is A330972.
The least number with n strict factorizations is A330974.
Cf. A001055, A001222, A033833, A045782, A318286, A325238, A330935, A330973, A330975.

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

A330975 Numbers that are not the number of factorizations of n into distinct factors > 1 for any n.

Original entry on oeis.org

11, 13, 20, 23, 24, 26, 28, 29, 30, 35, 36, 37, 39, 41, 45, 47, 48, 49, 50, 51, 53, 58, 60, 62, 63, 65, 66, 68, 69, 71, 72, 73, 75, 77, 78, 79, 81, 82, 84, 85, 86, 87, 90, 92, 94, 95, 96, 97, 98, 99, 101, 102, 103, 105, 106, 107, 108, 109, 113, 114, 115, 118

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

Warning: I have only confirmed the first three terms. The rest are derived from A045779. - Gus Wiseman, Jan 07 2020

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

Complement of A045779.
The non-strict version is A330976.
Factorizations are A001055, with image A045782, with complement A330976.
Strict factorizations are A045778, with image A045779.
The least positive integer with n strict factorizations is A330974(n).
Cf. A001222, A002033, A025487, A033833, A045780, A045783, A318286, A328966, A330972, A330973, A330997.

nn=20;
fam[n_]:=fam[n]=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[fam[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
nds=Length/@Array[Select[fam[#],UnsameQ@@#&]&,2^nn];
Complement[Range[nn],nds]

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

Original entry on oeis.org

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

cp's OEIS Frontend

A330991 Positive integers whose number of factorizations into factors > 1 (A001055) is a prime number (A000040).

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A045780 Least value with A045779(n) factorizations into distinct factors.

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A330974 Least positive integer with n factorizations into distinct factors > 1, and 0 if no such number exists.

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A330977 Numbers whose number of factorizations into factors > 1 (A001055) is a power of 2.

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A330997 Sorted list containing the least number with each possible nonzero number of factorizations into distinct factors > 1.

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A330975 Numbers that are not the number of factorizations of n into distinct factors > 1 for any n.

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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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