A330974 -id:A330974 - cp's OEIS frontend

A045783 Least value with A045782(n) factorizations.

1, 4, 8, 12, 16, 24, 36, 60, 48, 128, 72, 96, 120, 256, 180, 144, 192, 216, 420, 240, 1024, 384, 288, 360, 2048, 432, 480, 900, 768, 840, 576, 1260, 864, 720, 8192, 960, 1080, 1152, 4620, 1800, 3072, 1680, 1728, 1920, 1440, 32768, 2304, 2592, 6144

Offset: 1

David W. Wilson

nonn

			From _Gus Wiseman_, Jan 11 2020: (Start)
Factorizations of n = 1, 4, 8, 12, 16, 24, 36, 60, 48:
  {}  4    8      12     16       24       36       60       48
      2*2  2*4    2*6    2*8      3*8      4*9      2*30     6*8
           2*2*2  3*4    4*4      4*6      6*6      3*20     2*24
                  2*2*3  2*2*4    2*12     2*18     4*15     3*16
                         2*2*2*2  2*2*6    3*12     5*12     4*12
                                  2*3*4    2*2*9    6*10     2*3*8
                                  2*2*2*3  2*3*6    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
(End)

Wikipedia, Multiplicative partition
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.
The strict version is A045780.
The sorted version is A330972.
Includes all highly factorable numbers A033833.
The least number with exactly n factorizations is A330973(n).
Factorizations are A001055 with image A045782 and complement A330976.
Strict factorizations are A045778 with image A045779 and complement A330975.
Cf. A070175, A318284, A325238, A330974, A330989, A330992, A330998.

A330973 Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.

Original entry on oeis.org

1, 4, 8, 12, 16, 0, 24, 0, 36, 0, 60, 48, 0, 0, 128, 72, 0, 0, 96, 0, 120, 256, 0, 0, 0, 180, 0, 0, 144, 192, 216, 0, 0, 0, 0, 420, 0, 240, 0, 0, 0, 1024, 0, 0, 384, 0, 288, 0, 0, 0, 0, 360, 0, 0, 0, 2048, 432, 0, 0, 0, 0, 0, 0, 480, 0, 900, 768, 0, 0, 0, 0, 0

Offset: 1

Gus Wiseman, Jan 06 2020

nonn

Robert G. Wilson v, 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.

All nonzero terms belong to A025487.
Includes all highly factorable numbers A033833.
Factorizations are A001055, with image A045782.
The version without zeros is A045783.
The sorted version is A330972.
The strict version is A330974.
Positions of zeros are A330976.
Cf. A001055, A001222, A002033, A007716, A045778, A045779, A330935, A330992, A330997, A330998, A346426.

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[fam[#]&,2^nn];
Table[If[#=={},0,#[[1,1]]]&[Position[nds,i]],{i,nn}]

More terms from Jinyuan Wang, Jul 07 2021

A330976 Numbers that are not the number of factorizations into factors > 1 of any positive integer.

Original entry on oeis.org

6, 8, 10, 13, 14, 17, 18, 20, 23, 24, 25, 27, 28, 32, 33, 34, 35, 37, 39, 40, 41, 43, 44, 46, 48, 49, 50, 51, 53, 54, 55, 58, 59, 60, 61, 62, 63, 65, 68, 69, 70, 71, 72, 73, 75, 76, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 93, 94, 95, 96, 99

Offset: 1

Gus Wiseman, Jan 07 2020

nonn

Warning: I have only confirmed the first eight terms. The rest are derived from A045782. - 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 A045782.
The strict version is A330975.
Factorizations are A001055, with image A045782.
Strict factorizations are A045778, with image A045779.
The least number with n factorizations is A330973(n).
Cf. A033833, A045779, A045783, A317145, A330935, A330972, A330974, A330977, A330991.

nn=15;
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[fam[#]&,2^nn];
Complement[Range[nn],nds]

A045779 Number of factorizations of n into distinct factors for some n (image of A045778).

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 16, 17, 18, 19, 21, 22, 25, 27, 31, 32, 33, 34, 38, 40, 42, 43, 44, 46, 52, 54, 55, 56, 57, 59, 61, 64, 67, 70, 74, 76, 80, 83, 88, 89, 91, 93, 100, 104, 110, 111, 112, 116, 117, 120, 122, 123, 132, 137, 140, 141, 142, 143, 148

Offset: 1

David W. Wilson

nonn

We may use A045778(k*m) >= A045778(k) for any k, m >= 1 to disprove presence of some positive integer in this sequence. - David A. Corneth, Oct 24 2024

			From _David A. Corneth_, Oct 24 2024: (Start)
5 is a term as 24 has five factorizations into distinct divisors of 24 namely 24 = 2 * 12 = 3 * 8 = 4 * 6 = 2 * 3 * 4 which is five such factorizations.
11 is not a term. From terms in A025487 only the numbers 2, 4, 6, 8, 12, 16, 24, 30, 32, 36, 48, 60, 64, 72, 96, 128, 256, 512, 1024 have no more than 11 such factorizations. Any multiple of these numbers in A025487 that is not already listed has more than 11 such factorizations which proves 11 is not in this sequence. (End)

David A. Corneth, Table of n, a(n) for n = 1..953 (terms <= 10^5)
Wikipedia, Multiplicative partition
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.
Strict factorizations are A045778 with image A045779 and complement A330975.
The least number with A045779(n) strict factorizations is A045780(n).
The least number with n strict factorizations is A330974(n).
Cf. A001222, A025487, A033833, A045783, A318286, A328966, A330972, A330973, A330997.

Name edited by Gus Wiseman, Jan 11 2020

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

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]}]

A339661 Number of factorizations of n into distinct squarefree semiprimes.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 1, 1, 0, 1, 0, 1, 0, 0, 1, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 1, 1, 1, 0, 0, 1, 1, 0, 1, 1, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 1, 0

Offset: 1

Gus Wiseman, Dec 19 2020

nonn

A squarefree semiprime (A006881) is a product of any two distinct prime numbers.

Also the number of strict multiset partitions of the multiset of prime factors of n, into distinct strict pairs.

			The a(n) factorizations for n = 210, 1260, 4620, 30030, 69300 are respectively 3, 2, 6, 15, 7:
  (6*35)   (6*10*21)  (6*10*77)   (6*55*91)    (6*10*15*77)
  (10*21)  (6*14*15)  (6*14*55)   (6*65*77)    (6*10*21*55)
  (14*15)             (6*22*35)   (10*33*91)   (6*10*33*35)
                      (10*14*33)  (10*39*77)   (6*14*15*55)
                      (10*21*22)  (14*33*65)   (6*15*22*35)
                      (14*15*22)  (14*39*55)   (10*14*15*33)
                                  (15*22*91)   (10*15*21*22)
                                  (15*26*77)
                                  (21*22*65)
                                  (21*26*55)
                                  (22*35*39)
                                  (26*33*35)
                                  (6*35*143)
                                  (10*21*143)
                                  (14*15*143)

Antti Karttunen, Table of n, a(n) for n = 1..69300
Index entries for sequences computed from exponents in factorization of n

Dirichlet convolution of A008836 (Liouville's lambda) with A339742.
A050326 allows all squarefree numbers, non-strict case A050320.
A320656 is the not necessarily strict version.
A320911 lists all (not just distinct) products of squarefree semiprimes.
A322794 counts uniform factorizations, such as these.
A339561 lists positions of nonzero terms.
A001055 counts factorizations, with strict case A045778.
A001358 lists semiprimes, with squarefree case A006881.
A320655 counts factorizations into semiprimes, with strict case A322353.
The following count vertex-degree partitions and give their Heinz numbers:
- A000070 counts non-multigraphical partitions of 2n (A339620).
- A209816 counts multigraphical partitions (A320924).
- A339655 counts non-loop-graphical partitions of 2n (A339657).
- A339656 counts loop-graphical partitions (A339658).
- A339617 counts non-graphical partitions of 2n (A339618).
- A000569 counts graphical partitions (A320922).
The following count partitions of even length and give their Heinz numbers:
- A096373 cannot be partitioned into strict pairs (A320891).
- A338914 can be partitioned into strict pairs (A320911).
- A338915 cannot be partitioned into distinct pairs (A320892).
- A338916 can be partitioned into distinct pairs (A320912).
- A339559 cannot be partitioned into distinct strict pairs (A320894).
- A339560 can be partitioned into distinct strict pairs (A339561).
Cf. A001221, A005117, A007716, A028260, A280710, A300061, A320658, A320659, A320923, A330974.

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

A280710(n) = (bigomega(n)==2*issquarefree(n)); \\ From A280710.
A339661(n, u=(1+n)) = if(1==n, 1, my(s=0); fordiv(n, d, if((d>1) && (dA280710(d), s += A339661(n/d, d))); (s)); \\ Antti Karttunen, May 02 2022

a(n) = Sum_{d|n} (-1)^A001222(d) * A339742(n/d).

More terms and secondary offset added by Antti Karttunen, May 02 2022

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]

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

cp's OEIS Frontend

A045783 Least value with A045782(n) factorizations.

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A330973 Least positive integer with exactly n factorizations into factors > 1, and 0 if no such number exists.

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A330976 Numbers that are not the number of factorizations into factors > 1 of any positive integer.

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A045779 Number of factorizations of n into distinct factors for some n (image of A045778).

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

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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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A339661 Number of factorizations of n into distinct squarefree semiprimes.

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

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