A070039 -id:A070039 - cp's OEIS frontend

A341594 Number of strictly superior odd divisors of n.

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

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

			The a(n) divisors for n = 3, 15, 45, 105, 315, 405, 945, 1575, 1890:
  3  5   9   15   21   27   35   45    45
     15  15  21   35   45   45   63    63
         45  35   45   81   63   75    105
             105  63   135  105  105   135
                  105  405  135  175   189
                  315       189  225   315
                            315  315   945
                            945  525
                                 1575

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

On odd indices, equals A056924 (number of strictly superior divisors).
The inferior version is A069288.
Positions of zeros are A116882.
Positions of nonzero terms are A116883.
The strictly inferior version is A333805.
The version for squarefree instead of odd divisors is A341595.
The version for prime instead of odd divisors is A341642.
The version for prime-power instead of odd divisors is A341644.
The superior version is A341675.
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A140271 selects the smallest strictly superior divisor.
A207375 lists central divisors.
A341673 lists strictly superior divisors.
- Inferior: A063962, A066839, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591, A341592, A341593, A341676.
- Strictly Inferior: A060775, A070039, A333806, A341596, A341674.
- Strictly Superior: A048098, A064052, A238535, A341643, A341645, A341646.
Cf. A000005, A000203, A000265, A001222, A001248, A005408, A006530, A020639, A027193, A058695, A340101.

Table[Length[Select[Divisors[n],OddQ[#]&&#>n/#&]],{n,100}]

A341594(n) = sumdiv(n,d,(d%2)*(d>(n/d))); \\ Antti Karttunen, Feb 23 2021

A341596 Number of strictly inferior squarefree divisors of n.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 2, 2, 2, 1, 4, 1, 2, 2, 2, 2, 3, 1, 2, 2, 3, 1, 4, 1, 2, 3, 2, 1, 4, 1, 3, 2, 2, 1, 4, 2, 3, 2, 2, 1, 5, 1, 2, 3, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 2, 3, 2, 2, 4, 1, 3, 2, 2, 1, 5, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.

			The strictly inferior squarefree divisors of selected n:
n = 1  2  6  12  30  60  120  210  240  420  630  1050  1260
    --------------------------------------------------------
    {} 1  1  1   1   1   1    1    1    1    1    1     1
          2  2   2   2   2    2    2    2    2    2     2
             3   3   3   3    3    3    3    3    3     3
                 5   5   5    5    5    5    5    5     5
                     6   6    6    6    6    6    6     6
                         10   7    10   7    7    7     7
                              10   15   10   10   10    10
                              14        14   14   14    14
                                        15   15   15    15
                                             21   21    21
                                                  30    30
                                                        35

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of ones are A000430.
The weakly inferior version is A333749.
The version counting odd instead of squarefree divisors is A333805.
The version counting prime instead of squarefree divisors is A333806.
The weakly superior version is A341592.
The strictly superior version is A341595.
The version counting prime-power instead of squarefree divisors is A341677.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime power divisors.
A005117 lists squarefree numbers.
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A207375 lists central divisors.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A116882, A116883, A161908, A341591, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001055, A001248, A050320.

Table[Length[Select[Divisors[n],SquareFreeQ[#]&&#

a(n) = sumdiv(n, d, d^2 < n && issquarefree(d)); \\ Amiram Eldar, Nov 01 2024

A341646 Numbers with a strictly superior squarefree divisor.

Original entry on oeis.org

2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17, 18, 19, 20, 21, 22, 23, 24, 26, 28, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 50, 51, 52, 53, 55, 56, 57, 58, 59, 60, 61, 62, 63, 65, 66, 67, 68, 69, 70, 71, 73, 74, 75, 76, 77, 78, 79, 80

Offset: 1

Gus Wiseman, Feb 22 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.
This is a subsequence of A007916, i.e., no perfect powers appear here. [For perfect powers n, a supposed strictly superior squarefree divisor d=p*q*r... with distinct primes p,q,r,s...  has a complementary divisor n/d=p^i*q^j*r^k*s*... with i,j,k>=1, so the complementary divisor is at least as large as d, a contradiction.] Entries in A007916 but not in here are 48, 54, 72, 96, 108, 160, 162, 192,... - R. J. Mathar, Jul 07 2023
Is this a duplicate of A089105? - R. J. Mathar, Jul 24 2023

			60 has three strictly superior squarefree divisors {10,15,30} so 60 is in the sequence.

Amiram Eldar, Table of n, a(n) for n = 1..10000

The version for prime instead of squarefree divisors is A064052.
The version for prime-power instead of squarefree divisors is the complement of A051283.
The weakly superior version is the complement of A059172.
The version for odd instead of squarefree divisors is A116883.
These are the positions of nonzero terms in A341595.
The complement is A341645.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A140271 selects the smallest strictly superior divisor.
A207375 list central divisors.
A341673 lists strictly superior divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A033677, A063538, A063539, A070038, A072500, A116882, A161908, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly Superior: A048098, A238535, A341594, A341595, A341643, A341644.
Cf. A001222, A001248, A006530, A020639, A112798.
Subsequence of A007916.

isA341646 := proc(n)
    local d ;
    for d in numtheory[divisors](n) do
        if d>n/d then
            if issqrfree(d) then
                return true ;
            end if;
        end if;
    end do:
    false ;
end proc:
for n from 2 to 100 do
    if isA341646(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Jul 07 2023

Select[Range[100],Function[n,Select[Divisors[n],SquareFreeQ[#]&&#>n/#&]!={}]]

is(n) = fordiv(n, d, if(d^2 > n && issquarefree(d), return(1))); 0; \\ Amiram Eldar, Nov 01 2024

A341675 Number of superior odd divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

			The superior odd divisors of selected n:
n = 1  9  45  105  225  315  945  1575  1890  2835  3465  8190
   -----------------------------------------------------------
    1  9  45  105  225  315  945  1575  945   2835  3465  4095
       3  15  35   75   105  315  525   315   945   1155  1365
          9   21   45   63   189  315   189   567   693   819
              15   25   45   135  225   135   405   495   585
                   15   35   105  175   105   315   385   455
                        21   63   105   63    189   315   315
                             45   75    45    135   231   273
                             35   63          105   165   195
                                  45          81    105   117
                                              63    99    105
                                                    77    91
                                                    63

Antti Karttunen, Table of n, a(n) for n = 1..20000

The inferior version is A069288.
Positions of 0's (except a(1)) are A116882.
The version for prime instead of odd divisors is A341591.
The version for squarefree instead of odd divisors is A341592.
The version for prime-power instead of odd divisors is A341593.
The strictly superior case is A341594.
A001221 counts prime divisors, with sum A001414.
A005117 lists squarefree numbers.
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A161908 lists superior divisors.
A207375 lists central divisors.
- Inferior: A063962, A066839, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A116882, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly Superior: A048098, A064052 A140271, A238535, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001222, A001248, A006530, A020639.

Table[Length[Select[Divisors[n],OddQ[#]&&#>=n/#&]],{n,100}]

A341675(n) = sumdiv(n,d,(d%2)*(d>=n/d)); \\ Antti Karttunen, Dec 13 2021

Data section extended up to 105 terms by Antti Karttunen, Dec 13 2021

A341676 The unique superior prime divisor of each number that has one.

Original entry on oeis.org

2, 3, 2, 5, 3, 7, 3, 5, 11, 13, 7, 5, 17, 19, 5, 7, 11, 23, 5, 13, 7, 29, 31, 11, 17, 7, 37, 19, 13, 41, 7, 43, 11, 23, 47, 7, 17, 13, 53, 11, 19, 29, 59, 61, 31, 13, 11, 67, 17, 23, 71, 73, 37, 19, 11, 13, 79, 41, 83, 17, 43, 29, 11, 89, 13, 23, 31, 47, 19

Offset: 1

Gus Wiseman, Feb 23 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908. Numbers with a superior prime divisor are listed by A063538.

			The sequence of superior prime divisors begins: {}, {2}, {3}, {2}, {5}, {3}, {7}, {}, {3}, {5}, {11}, {}, {13}, {7}, {5}, {}, {17}, {}, {19}, {5}, ...

Amiram Eldar, Table of n, a(n) for n = 1..10000

Inferior versions are A107286 (smallest), A217581 (largest), A056608.
These divisors (superior prime) are counted by A341591.
The strictly superior version is A341643.
A001221 counts prime divisors, with sum A001414.
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A060775 selects the greatest strictly inferior divisor.
A063538/A063539 have/lack a superior prime divisor.
A070038 adds up superior divisors.
A140271 selects the smallest strictly superior divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
- Inferior: A063962, A066839, A069288, A161906, A333749, A333750.
- Superior: A051283, A059172, A063539, A070038, A116882, A341592, A341593.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798.

Join@@Table[Select[Divisors[n],PrimeQ[#]&&#>=n/#&],{n,100}]

lista(nmax) = {my(p); for(n = 1, nmax, p = select(x -> (x^2 >= n), factor(n)[, 1]); if(#p == 1, print1(p[1], ", ")));} \\ Amiram Eldar, Nov 01 2024

A341645 Numbers without a strictly superior squarefree divisor.

Original entry on oeis.org

1, 4, 8, 9, 16, 25, 27, 32, 36, 48, 49, 54, 64, 72, 81, 96, 100, 108, 121, 125, 128, 144, 160, 162, 169, 192, 196, 200, 216, 224, 225, 243, 250, 256, 288, 289, 320, 324, 343, 361, 375, 384, 392, 400, 405, 432, 441, 448, 484, 486, 500, 512, 529, 567, 576, 625

Offset: 1

Gus Wiseman, Feb 22 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

			72 has strictly superior divisors {9,12,18,24,36,72} and squarefree divisors {1,2,3,6}, but the intersection is empty, so 72 is in the sequence.

Michael De Vlieger, Table of n, a(n) for n = 1..10000

The version for prime instead of squarefree divisors is A048098.
The version for prime-power instead of squarefree divisors is A051283.
The weakly superior version is A059172.
The version for odd instead of squarefree divisors is A116882.
Positions of zeros in A341595.
The complement is A341646.
A001221 counts prime divisors, with sum A001414.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A140271 selects the smallest strictly superior divisor.
A207375 list central divisors.
A341673 lists strictly superior divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A033677, A063538, A063539, A070038, A072500, A116883, A161908, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly Superior: A064052, A238535, A341594, A341642, A341643, A341644.
Cf. A000005, A000203, A001222, A001248, A006530, A020639, A112798.

Select[Range[100],Function[n,Select[Divisors[n],SquareFreeQ[#]&&#>n/#&]=={}]]

isok(m) = my(d=divisors(m)); #select(x->(issquarefree(x) && (x^2>m)), d) == 0; \\ Michel Marcus, Feb 11 2024

A341593 Number of superior prime-power divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 19 2021

nonn

We define a divisor d|n to be superior if d >= n/d. Superior divisors are counted by A038548 and listed by A161908.

			The superior prime-power divisors (columns) of selected n:
n = 4374  5103  6144  7500  9000
    ----------------------------
      81    81   128   125   125
     243   243   256   625
     729   729   512
    2187        1024
                2048

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of zeros after the first are A051283.
The inferior version is A333750.
The version for prime instead of prime-power divisors is A341591.
The version for squarefree instead of prime-power divisors is A341592.
Dominates A341644 (the strictly superior case).
The version for odd instead of prime-power divisors is A341675.
The strictly inferior version is A341677.
A000961 lists prime powers.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A161908 lists superior divisors.
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A059172, A063538, A063539, A070038, A116882, A116883, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly Superior: A048098, A064052 A140271, A238535, A341594, A341595, A341642, A341643, A341645, A341646, A341673.
Cf. A000005, A000203, A001248, A006530, A020639, A112798.

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#>=n/#&]],{n,100}]

a(n) = sumdiv(n, d, d^2 >= n && isprimepower(d)); \\ Amiram Eldar, Nov 01 2024

A341644 Number of strictly superior prime-power divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 22 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.

			The strictly superior prime power divisors of random selected n:
n = 768  2048  5103  6144  8192  8722  9433  9984
   ----------------------------------------------
     32    64    81   128   128        9433  128
     64   128   243   256   256              256
    128   256   729   512   512
    256   512        1024  1024
         1024        2048  2048
         2048              4096
                           8192

Amiram Eldar, Table of n, a(n) for n = 1..10000

Positions of zeros (after the first) are A051283.
The inferior version is A333750.
Dominated by A341593 (the weakly superior version).
The version for odd instead of prime divisors is A341594.
The version for squarefree instead of prime divisors is A341595.
The version for prime instead of prime-power divisors is A341642.
The strictly inferior version is A341677.
A000961 lists prime powers.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A140271 selects the smallest strictly superior divisor.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A207375 list central divisors.
A341673 lists strictly superior divisors.
- Inferior: A033676, A063962, A066839, A069288, A161906, A217581, A333749.
- Superior: A033677, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A161908, A341591, A341592, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly Superior: A048098, A064052, A238535, A341643, A341646.
Cf. A000005, A000203, A001248, A006530, A020639, A112798.

Table[Length[Select[Divisors[n],PrimePowerQ[#]&&#>n/#&]],{n,100}]

a(n) = sumdiv(n, d, d^2 > n && isprimepower(d)); \\ Amiram Eldar, Nov 01 2024

A342083 Number of chains of strictly inferior divisors from n to 1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 3, 1, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 5, 1, 3, 3, 2, 1, 6, 1, 3, 2, 3, 1, 5, 2, 4, 2, 2, 1, 7, 1, 2, 3, 3, 2, 5, 1, 3, 2, 4, 1, 8, 1, 2, 3, 3, 2, 5, 1, 6, 2, 2, 1, 7, 2, 2, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly inferior if d < n/d. Strictly inferior divisors are counted by A056924 and listed by A341674.
These chains have first-quotients (in analogy with first-differences) that are term-wise > their decapitation (maximum element removed). Equivalently, x > y^2 for all adjacent x, y. For example, the divisor chain q = 60/6/2/1 has first-quotients (10,3,2), which are > (6,2,1), so q is counted under a(60).
Also the number of factorizations of n where each factor is greater than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2/1  6/1    12/1    24/1    42/1      48/1      60/1      72/1
       6/2/1  12/2/1  24/2/1  42/2/1    48/2/1    60/2/1    72/2/1
              12/3/1  24/3/1  42/3/1    48/3/1    60/3/1    72/3/1
                      24/4/1  42/6/1    48/4/1    60/4/1    72/4/1
                              42/6/2/1  48/6/1    60/5/1    72/6/1
                                        48/6/2/1  60/6/1    72/8/1
                                                  60/6/2/1  72/6/2/1
                                                            72/8/2/1
The a(n) factorizations for n = 2, 6, 12, 24, 42, 48, 60, 72:
  2  6    12   24    42     48     60      72
     2*3  2*6  3*8   6*7    6*8    2*30    8*9
          3*4  4*6   2*21   2*24   3*20    2*36
               2*12  3*14   3*16   4*15    3*24
                     2*3*7  4*12   5*12    4*18
                            2*3*8  6*10    6*12
                                   2*3*10  2*4*9
                                           2*3*12

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A040039.
Not requiring strict inferiority gives A074206 (ordered factorizations).
The weakly inferior version is A337135.
The strictly superior version is A342084.
The weakly superior version is A342085.
The additive version is A342098, or A000929 allowing equality.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts chains of divisors summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
A342086 counts chains of divisors with strictly increasing quotients > 1.
- Inferior: A033676, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A048098, A064052, A140271, A238535, A341673.
Cf. A000203, A001248, A002033, A006530, A018819, A020639, A045690, A337105, A342087, A342094, A342095, A342096, A342097.

cen[n_]:=If[n==1,{{1}},Prepend[#,n]&/@Join@@cen/@Select[Divisors[n],#

G.f.: x + Sum_{k>=1} a(k) * x^(k*(k + 1)) / (1 - x^k). - Ilya Gutkovskiy, Nov 03 2021

A342084 Number of chains of distinct strictly superior divisors starting with n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 2, 1, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 4, 2, 2, 1, 9, 1, 2, 2, 4, 1, 7, 1, 6, 2, 2, 2, 10, 1, 2, 2, 9, 1, 6, 1, 4, 4, 2, 1, 19, 1, 4, 2, 4, 1, 8, 2, 9, 2, 2, 1, 20, 1, 2, 4, 10, 2, 6, 1, 4, 2, 7, 1, 29, 1, 2, 4, 4, 2, 6, 1, 19, 3, 2, 1, 19, 2

Offset: 1

Gus Wiseman, Feb 28 2021

nonn

We define a divisor d|n to be strictly superior if d > n/d. Strictly superior divisors are counted by A056924 and listed by A341673.
These chains have first-quotients (in analogy with first-differences) that are term-wise < their decapitation (maximum element removed). Equivalently, x < y^2 for all adjacent x, y. For example, the divisor chain q = 30/6/3 has first-quotients (5,2), which are < (6,3), so q is counted under a(30).
Also the number of ordered factorizations of n where each factor is less than the product of all previous factors.

			The a(n) chains for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12      16      24         30       32         36
     6/3  12/4    16/8    24/6       30/6     32/8       36/9
          12/6    16/8/4  24/8       30/10    32/16      36/12
          12/6/3          24/12      30/15    32/8/4     36/18
                          24/6/3     30/6/3   32/16/8    36/12/4
                          24/8/4     30/10/5  32/16/8/4  36/12/6
                          24/12/4    30/15/5             36/18/6
                          24/12/6                        36/18/9
                          24/12/6/3                      36/12/6/3
                                                         36/18/6/3
The a(n) ordered factorizations for n = 2, 6, 12, 16, 24, 30, 32, 36:
  2  6    12     16     24       30     32       36
     3*2  4*3    8*2    6*4      6*5    8*4      9*4
          6*2    4*2*2  8*3      10*3   16*2     12*3
          3*2*2         12*2     15*2   4*2*4    18*2
                        3*2*4    3*2*5  8*2*2    4*3*3
                        4*2*3    5*2*3  4*2*2*2  6*2*3
                        4*3*2    5*3*2           6*3*2
                        6*2*2                    9*2*2
                        3*2*2*2                  3*2*2*3
                                                 3*2*3*2

Amiram Eldar, Table of n, a(n) for n = 1..10000

The restriction to powers of 2 is A045690, with reciprocal version A040039.
The inferior version is A337135.
The strictly inferior version is A342083.
The superior version is A342085.
The additive version allowing equality is A342094 or A342095.
The additive version is A342096 or A342097.
A000005 counts divisors.
A001055 counts factorizations.
A003238 counts divisibility chains summing to n-1, with strict case A122651.
A038548 counts inferior (or superior) divisors.
A056924 counts strictly inferior (or strictly superior) divisors.
A067824 counts strict chains of divisors starting with n.
A074206 counts strict chains of divisors from n to 1 (also ordered factorizations).
A167865 counts strict chains of divisors > 1 summing to n.
A207375 lists central divisors.
A253249 counts strict chains of divisors.
A334996 counts ordered factorizations by product and length.
A334997 counts chains of divisors of n by length.
- Inferior: A033676, A063962, A066839, A072499, A161906.
- Superior: A033677, A070038, A161908, A341591.
- Strictly Inferior: A060775, A070039, A333806, A341674.
- Strictly Superior: A064052/A048098, A140271, A238535, A341642, A341673.
Cf. A000203, A000929, A001248, A006530, A018819, A020639, A169594, A337105, A342098.

ceo[n_]:=Prepend[Prepend[#,n]&/@Join@@ceo/@Select[Most[Divisors[n]],#>n/#&],{n}];
Table[Length[ceo[n]],{n,100}]

a(2^n) = A045690(n).

cp's OEIS Frontend

A341594 Number of strictly superior odd divisors of n.

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A341596 Number of strictly inferior squarefree divisors of n.

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A341646 Numbers with a strictly superior squarefree divisor.

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A341675 Number of superior odd divisors of n.

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A341676 The unique superior prime divisor of each number that has one.

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A341645 Numbers without a strictly superior squarefree divisor.

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A341593 Number of superior prime-power divisors of n.

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