A341677 -id:A341677 - cp's OEIS frontend

A341596 Number of strictly inferior squarefree divisors of n.

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

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

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

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A341596 Number of strictly inferior squarefree divisors of n.

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

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A341593 Number of superior prime-power divisors of n.

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

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