A340597 -id:A340597 - 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

A340596 Number of co-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 4, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 5, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 3, 1, 4, 1, 2, 1, 1, 1, 5, 1, 2, 2, 4, 1, 1, 1, 3, 1, 1, 1, 5, 1, 1, 1, 4, 1, 1, 1, 2, 2, 1, 1, 8

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be co-balanced if it has exactly A001221(n) factors.

			The a(n) co-balanced factorizations for n = 12, 24, 36, 72, 120, 144, 180:
  2*6    3*8     4*9     8*9     3*5*8     2*72     4*5*9
  3*4    4*6     6*6     2*36    4*5*6     3*48     5*6*6
         2*12    2*18    3*24    2*2*30    4*36     2*2*45
                 3*12    4*18    2*3*20    6*24     2*3*30
                         6*12    2*4*15    8*18     2*5*18
                                 2*5*12    9*16     2*6*15
                                 2*6*10    12*12    2*9*10
                                 3*4*10             3*3*20
                                                    3*4*15
                                                    3*5*12
                                                    3*6*10

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The version for unlabeled multiset partitions is A319616.
The alt-balanced version is A340599.
The balanced version is A340653.
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.
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 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A006141, A050320, A112798, A117409, A324518, A339846, A339890, A340607, A340656, A340657.

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[#]==PrimeNu[n]&]],{n,100}]

A340596(n, m=n, om=omega(n)) = if(1==n,(0==om), sumdiv(n, d, if((d>1)&&(d<=m), A340596(n/d, d, om-1)))); \\ Antti Karttunen, Jun 10 2024

Data section extended up to a(120) by Antti Karttunen, Jun 10 2024

A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 20 2021

nonn

I call these alt-balanced factorizations. Balanced factorizations are A340653. - Gus Wiseman, Jan 20 2021

			The alt-balanced factorizations for n = 192, 1728, 3456, 9216:
  3*4*4*4       2*2*2*6*6*6   2*2*4*6*6*6         4*4*4*4*6*6
  2*2*2*2*2*6   2*2*3*4*6*6   2*3*4*4*6*6         2*2*2*2*2*6*6*8
                2*3*3*4*4*6   3*3*4*4*4*6         2*2*2*2*3*3*8*8
                              2*2*2*2*3*3*3*8     2*2*2*2*3*4*6*8
                              2*2*2*2*2*2*2*3*9   2*2*2*3*3*4*4*8
                                                  2*2*2*2*2*2*2*8*9
                                                  2*2*2*2*2*2*4*4*9

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

The co-balanced version is A340596.
Positions of nonzero terms are A340597.
The case of powers of two is A340611.
Taking maximum Omega instead of maximum factor gives A340653.
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.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340598 counts balanced set partitions.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A006141, A117409, A320655, A320656, A324518, A339846, A339890, A340607.

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[#]&]],{n,100}]

A340599(n, m=n, e=0, mf=1) = if(1==n, mf==e, sumdiv(n, d, if((d>1)&&(d<=m), A340599(n/d, d, 1+e, max(d, mf))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(120) and the secondary offset added by Antti Karttunen, Jun 19 2024

A340654 Number of cross-balanced factorizations of n.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 1, 3, 1, 1, 1, 2, 1, 1, 1, 1, 1, 1, 1, 5, 1, 1, 1, 3, 1, 1, 1, 2, 2, 1, 1, 4, 1, 2, 1, 2, 1, 3, 1, 3, 1, 1, 1, 3, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 1, 6, 1, 1, 2, 2, 1, 1, 1, 4, 1, 1, 1, 3, 1, 1, 1, 3, 1, 3, 1, 2, 1, 1, 1, 5, 1, 2, 2, 5, 1, 1, 1, 3, 1

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be cross-balanced if either (1) it is empty or (2) the maximum image of A001222 over the factors is A001221(n).

			The cross-balanced factorizations for n = 12, 24, 36, 72, 144, 240:
  2*6   4*6     4*9     2*4*9     4*4*9       8*30
  3*4   2*2*6   6*6     2*6*6     4*6*6       12*20
        2*3*4   2*2*9   3*4*6     2*2*4*9     5*6*8
                2*3*6   2*2*2*9   2*2*6*6     2*4*30
                3*3*4   2*2*3*6   2*3*4*6     2*6*20
                        2*3*3*4   3*3*4*4     2*8*15
                                  2*2*2*2*9   3*4*20
                                  2*2*2*3*6   3*8*10
                                  2*2*3*3*4   4*5*12
                                              2*10*12
                                              2*3*5*8
                                              2*2*2*30
                                              2*2*3*20
                                              2*2*5*12

Antti Karttunen, Table of n, a(n) for n = 1..65537

Positions of terms > 1 are A126706.
Positions of 1's are A303554.
The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340651.
The balanced version is A340653.
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.
- A340599 counts alt-balanced factorizations.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340656 have no twice-balanced factorizations.
- A340657 have a twice-balanced factorization.
Cf. A003963, A117409, A303975, A320656, A324518, 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],#=={}||PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,100}]

A340654(n, m=n, om=omega(n),mbo=0) = if(1==n,(mbo==om), sumdiv(n, d, if((d>1)&&(d<=m), A340654(n/d, d, om, max(mbo,bigomega(d)))))); \\ Antti Karttunen, Jun 19 2024

Data section extended up to a(105) by Antti Karttunen, Jun 19 2024

A340655 Number of twice-balanced factorizations of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Jan 15 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The twice-balanced factorizations for n = 12, 120, 360, 480, 900, 2520:
  2*6   3*5*8    5*8*9     2*8*30    2*6*75    2*2*7*90
  3*4   2*2*30   2*4*45    3*8*20    2*9*50    2*3*5*84
        2*3*20   2*6*30    4*4*30    3*4*75    2*3*7*60
        2*5*12   2*9*20    4*6*20    3*6*50    2*5*7*36
                 3*4*30    4*8*15    4*5*45    3*3*5*56
                 3*6*20    5*8*12    5*6*30    3*3*7*40
                 3*8*15    6*8*10    5*9*20    3*5*7*24
                 4*5*18    2*12*20   2*10*45   2*2*2*315
                 5*6*12    4*10*12   2*15*30   2*2*3*210
                 2*10*18             2*18*25   2*2*5*126
                 2*12*15             3*10*30   2*3*3*140
                 3*10*12             3*12*25
                                     3*15*20
                                     5*10*18
                                     5*12*15

The co-balanced version is A340596.
The version for unlabeled multiset partitions is A340652.
The balanced version is A340653.
The cross-balanced version is A340654.
Positions of zeros are A340656.
Positions of nonzero terms are A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
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.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
Cf. A003963, A050336, A117409, A320655, A320656, A324518, A339846, A339890, A340607, 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[#]==PrimeNu[n]==Max[PrimeOmega/@#]&]],{n,30}]

A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

1, 2, 4, 6, 8, 10, 14, 16, 20, 22, 26, 28, 32, 34, 38, 44, 46, 52, 58, 62, 64, 68, 74, 76, 82, 86, 88, 92, 94, 104, 106, 116, 118, 122, 124, 128, 134, 136, 142, 146, 148, 152, 158, 164, 166, 172, 178, 184, 188, 194, 202, 206, 212, 214, 218, 226, 232, 236, 244

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Consists of 1 and all numbers that are even and have no odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
      1: {}              44: {1,1,5}          106: {1,16}
      2: {1}             46: {1,9}            116: {1,1,10}
      4: {1,1}           52: {1,1,6}          118: {1,17}
      6: {1,2}           58: {1,10}           122: {1,18}
      8: {1,1,1}         62: {1,11}           124: {1,1,11}
     10: {1,3}           64: {1,1,1,1,1,1}    128: {1,1,1,1,1,1,1}
     14: {1,4}           68: {1,1,7}          134: {1,19}
     16: {1,1,1,1}       74: {1,12}           136: {1,1,1,7}
     20: {1,1,3}         76: {1,1,8}          142: {1,20}
     22: {1,5}           82: {1,13}           146: {1,21}
     26: {1,6}           86: {1,14}           148: {1,1,12}
     28: {1,1,4}         88: {1,1,1,5}        152: {1,1,1,8}
     32: {1,1,1,1,1}     92: {1,1,9}          158: {1,22}
     34: {1,7}           94: {1,15}           164: {1,1,13}
     38: {1,8}          104: {1,1,1,6}        166: {1,23}
For example, the factorizations of 88 are (2*2*2*11), (2*2*22), (2*4*11), (2*44), (4*22), (8*11), (88), none of which has odd minimum, so 88 is in the sequence.

The version looking at greatest factor is A000079.
The version for twice-balanced is A340656, with complement A340657.
These factorization are counted by A340832.
The complement is A340855.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n==1||EvenQ[n]&&Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]=={}]]

A340855 Numbers that can be factored into factors > 1, the least of which is odd.

Original entry on oeis.org

3, 5, 7, 9, 11, 12, 13, 15, 17, 18, 19, 21, 23, 24, 25, 27, 29, 30, 31, 33, 35, 36, 37, 39, 40, 41, 42, 43, 45, 47, 48, 49, 50, 51, 53, 54, 55, 56, 57, 59, 60, 61, 63, 65, 66, 67, 69, 70, 71, 72, 73, 75, 77, 78, 79, 80, 81, 83, 84, 85, 87, 89, 90, 91, 93, 95

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

These are numbers that are odd or have an odd divisor 1 < d <= n/d.

			The sequence of terms together with their prime indices begins:
     3: {2}          27: {2,2,2}      48: {1,1,1,1,2}
     5: {3}          29: {10}         49: {4,4}
     7: {4}          30: {1,2,3}      50: {1,3,3}
     9: {2,2}        31: {11}         51: {2,7}
    11: {5}          33: {2,5}        53: {16}
    12: {1,1,2}      35: {3,4}        54: {1,2,2,2}
    13: {6}          36: {1,1,2,2}    55: {3,5}
    15: {2,3}        37: {12}         56: {1,1,1,4}
    17: {7}          39: {2,6}        57: {2,8}
    18: {1,2,2}      40: {1,1,1,3}    59: {17}
    19: {8}          41: {13}         60: {1,1,2,3}
    21: {2,4}        42: {1,2,4}      61: {18}
    23: {9}          43: {14}         63: {2,2,4}
    24: {1,1,1,2}    45: {2,2,3}      65: {3,6}
    25: {3,3}        47: {15}         66: {1,2,5}
For example, 72 is in the sequence because it has three suitable factorizations: (3*3*8), (3*4*6), (3*24).

The version looking at greatest factor is A057716.
The version for twice-balanced is A340657, with complement A340656.
These factorization are counted by A340832.
The complement is A340854.
A033676 selects the maximum inferior divisor.
A038548 counts inferior divisors, listed by A161906.
A055396 selects the least prime index.
- Factorizations -
A001055 counts factorizations.
A045778 counts strict factorizations.
A316439 counts factorizations by product and length.
A339890 counts factorizations of odd length.
A340653 counts balanced factorizations.
- Odd -
A000009 counts partitions into odd parts.
A024429 counts set partitions of odd length.
A026424 lists numbers with odd Omega.
A066208 lists Heinz numbers of partitions into odd parts.
A067659 counts strict partitions of odd length (A030059).
A174726 counts ordered factorizations of odd length.
A332304 counts strict compositions of odd length.
A340692 counts partitions of odd rank.
Cf. A026804, A027193, A050320, A244991, A340101, A340102, A340596, A340597, A340607, A340654, A340655, A340852.

Select[Range[100],Function[n,n>1&&(OddQ[n]||Select[Rest[Divisors[n]],OddQ[#]&&#<=n/#&]!={})]]

A340656 Numbers without a twice-balanced factorization.

Original entry on oeis.org

4, 6, 8, 9, 10, 14, 15, 16, 21, 22, 25, 26, 27, 30, 32, 33, 34, 35, 38, 39, 42, 46, 48, 49, 51, 55, 57, 58, 60, 62, 64, 65, 66, 69, 70, 72, 74, 77, 78, 80, 81, 82, 84, 85, 86, 87, 90, 91, 93, 94, 95, 96, 102, 105, 106, 108, 110, 111, 112, 114, 115, 118, 119

Offset: 1

Gus Wiseman, Jan 16 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The sequence of terms together with their prime indices begins:
     4: {1,1}          33: {2,5}          64: {1,1,1,1,1,1}
     6: {1,2}          34: {1,7}          65: {3,6}
     8: {1,1,1}        35: {3,4}          66: {1,2,5}
     9: {2,2}          38: {1,8}          69: {2,9}
    10: {1,3}          39: {2,6}          70: {1,3,4}
    14: {1,4}          42: {1,2,4}        72: {1,1,1,2,2}
    15: {2,3}          46: {1,9}          74: {1,12}
    16: {1,1,1,1}      48: {1,1,1,1,2}    77: {4,5}
    21: {2,4}          49: {4,4}          78: {1,2,6}
    22: {1,5}          51: {2,7}          80: {1,1,1,1,3}
    25: {3,3}          55: {3,5}          81: {2,2,2,2}
    26: {1,6}          57: {2,8}          82: {1,13}
    27: {2,2,2}        58: {1,10}         84: {1,1,2,4}
    30: {1,2,3}        60: {1,1,2,3}      85: {3,7}
    32: {1,1,1,1,1}    62: {1,11}         86: {1,14}
For example, the factorizations of 48 with (2) and (3) equal are: (2*2*2*6), (2*2*3*4), (2*4*6), (3*4*4), but since none of these has length 2, the sequence contains 48.

Positions of zeros in A340655.
The complement is A340657.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340597 lists numbers with an alt-balanced factorization.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A112798, A117409, A325134, A339846, A339890, A340607, A340689, A340690.

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

A340657 Numbers with a twice-balanced factorization.

Original entry on oeis.org

1, 2, 3, 5, 7, 11, 12, 13, 17, 18, 19, 20, 23, 24, 28, 29, 31, 36, 37, 40, 41, 43, 44, 45, 47, 50, 52, 53, 54, 56, 59, 61, 63, 67, 68, 71, 73, 75, 76, 79, 83, 88, 89, 92, 97, 98, 99, 100, 101, 103, 104, 107, 109, 113, 116, 117, 120, 124, 127, 131, 135, 136, 137

Offset: 1

Gus Wiseman, Jan 17 2021

nonn

We define a factorization of n into factors > 1 to be twice-balanced if it is empty or the following are equal:
(1) the number of factors;
(2) the maximum image of A001222 over the factors;
(3) A001221(n).

			The sequence of terms together with their prime indices begins:
      1: {}            29: {10}          59: {17}
      2: {1}           31: {11}          61: {18}
      3: {2}           36: {1,1,2,2}     63: {2,2,4}
      5: {3}           37: {12}          67: {19}
      7: {4}           40: {1,1,1,3}     68: {1,1,7}
     11: {5}           41: {13}          71: {20}
     12: {1,1,2}       43: {14}          73: {21}
     13: {6}           44: {1,1,5}       75: {2,3,3}
     17: {7}           45: {2,2,3}       76: {1,1,8}
     18: {1,2,2}       47: {15}          79: {22}
     19: {8}           50: {1,3,3}       83: {23}
     20: {1,1,3}       52: {1,1,6}       88: {1,1,1,5}
     23: {9}           53: {16}          89: {24}
     24: {1,1,1,2}     54: {1,2,2,2}     92: {1,1,9}
     28: {1,1,4}       56: {1,1,1,4}     97: {25}
The twice-balanced factorizations of 1920 (with prime indices {1,1,1,1,1,1,1,2,3}) are (8*8*30) and (8*12*20), so 1920 is in the sequence.

The alt-balanced version is A340597.
Positions of nonzero terms in A340655.
The complement is A340656.
A001055 counts factorizations.
A001221 counts distinct prime factors.
A001222 counts prime factors with multiplicity.
A045778 counts strict factorizations.
A303975 counts distinct prime factors in prime indices.
A316439 counts factorizations by product and length.
Other balance-related sequences:
- A010054 counts balanced strict partitions.
- A047993 counts balanced partitions.
- A098124 counts balanced compositions.
- A106529 lists Heinz numbers of balanced partitions.
- A340596 counts co-balanced factorizations.
- A340598 counts balanced set partitions.
- A340599 counts alt-balanced factorizations.
- A340600 counts unlabeled balanced multiset partitions.
- A340652 counts unlabeled twice-balanced multiset partitions.
- A340653 counts balanced factorizations.
- A340654 counts cross-balanced factorizations.
Cf. A005117, A056239, A112798, A117409, A320325, A325134, A339846, A339890, A340607, A340689, A340690.

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

A340852 Numbers that can be factored in such a way that every factor is a divisor of the number of factors.

Original entry on oeis.org

1, 4, 16, 27, 32, 64, 96, 128, 144, 192, 216, 256, 288, 324, 432, 486, 512, 576, 648, 729, 864, 972, 1024, 1296, 1458, 1728, 1944, 2048, 2560, 2592, 2916, 3125, 3888, 4096, 5120, 5184, 5832, 6144, 6400, 7776, 8192, 9216, 11664, 12288, 12800, 13824, 15552

Offset: 1

Gus Wiseman, Feb 04 2021

nonn

Also numbers that can be factored in such a way that the length is divisible by the least common multiple.

			The sequence of terms together with their prime indices begins:
    1: {}
    4: {1,1}
   16: {1,1,1,1}
   27: {2,2,2}
   32: {1,1,1,1,1}
   64: {1,1,1,1,1,1}
   96: {1,1,1,1,1,2}
  128: {1,1,1,1,1,1,1}
  144: {1,1,1,1,2,2}
  192: {1,1,1,1,1,1,2}
  216: {1,1,1,2,2,2}
  256: {1,1,1,1,1,1,1,1}
  288: {1,1,1,1,1,2,2}
  324: {1,1,2,2,2,2}
  432: {1,1,1,1,2,2,2}
For example, 24576 has three suitable factorizations:
  (2*2*2*2*2*2*2*2*2*2*2*12)
  (2*2*2*2*2*2*2*2*2*2*4*6)
  (2*2*2*2*2*2*2*2*2*3*4*4)
so is in the sequence.

Partitions of this type are counted by A340693 (A340606).
These factorizations are counted by A340851.
The reciprocal version is A340853.
A143773 counts partitions whose parts are multiples of the number of parts.
A320911 can be factored into squarefree semiprimes.
A340597 have an alt-balanced factorization.
A340656 lack a twice-balanced factorization, complement A340657.
- Factorizations -
A001055 counts factorizations, with strict case A045778.
A316439 counts factorizations by product and length.
A339846 counts factorizations of even length.
A339890 counts factorizations of odd length.
A340101 counts factorizations into odd factors, odd-length case A340102.
A340653 counts balanced factorizations.
A340831/A340832 count factorizations with odd maximum/minimum.
A340785 counts factorizations into even numbers, even-length case A340786.
A340854 cannot be factored with odd least factor, complement A340855.
Cf. A050320, A067538, A168659, A301987, A316413, A327517, A340596, A340599, A340609, A340654, A340655, A340827, A340830.

facs[n_]:=If[n<=1,{{}},Join@@Table[Map[Prepend[#,d]&,Select[facs[n/d],Min@@#>=d&]],{d,Rest[Divisors[n]]}]];
Select[Range[1000],Select[facs[#],And@@IntegerQ/@(Length[#]/#)&]!={}&]

cp's OEIS Frontend

A340653 Number of balanced factorizations of n.

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A340596 Number of co-balanced factorizations of n.

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A340599 Number of factorizations of n into factors > 1 with length and greatest factor equal.

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A340654 Number of cross-balanced factorizations of n.

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A340655 Number of twice-balanced factorizations of n.

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A340854 Numbers that cannot be factored into factors > 1, the least of which is odd.

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A340855 Numbers that can be factored into factors > 1, the least of which is odd.

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A340656 Numbers without a twice-balanced factorization.

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