A341677 -id:A341677 - cp's OEIS frontend

A033677 Smallest divisor of n >= sqrt(n).

1, 2, 3, 2, 5, 3, 7, 4, 3, 5, 11, 4, 13, 7, 5, 4, 17, 6, 19, 5, 7, 11, 23, 6, 5, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 6, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 7, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 8, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10

Offset: 1

N. J. A. Sloane

a(n) is the smallest k such that n appears in the k X k multiplication table and A027424(k) is the number of n with a(n) <= k.
a(n) is the largest central divisor of n. Right border of A207375. - Omar E. Pol, Feb 26 2019
If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence selects the smallest superior divisor of n. - Gus Wiseman, Feb 19 2021
a(p) = p for p a prime or 1, these are also the record high points in this sequence. - Charles Kusniec, Aug 26 2022
a(n^4+n^2+1) = n^2+n+1 (see A033676). - Jianing Song, Oct 23 2022

			From _Gus Wiseman_, Feb 19 2021: (Start)
The divisors of 36 are {1,2,3,4,6,9,12,18,36}. Of these {1,2,3,4,6} are inferior and {6,9,12,18,36} are superior, so a(36) = 6.
The divisors of 40 are {1,2,4,5,8,10,20,40}. Of these {1,2,4,5} are inferior and {8,10,20,40} are superior, so a(40) = 8.
(End)

G. Tenenbaum, pp. 268ff of R. L. Graham et al., eds., Mathematics of Paul Erdős I.

T. D. Noe, Table of n, a(n) for n = 1..10000

Cf. A027424, A056737, A060775, A219695.
The lower central divisor is A033676.
The strictly superior case is A140271.
Leftmost column of A161908 (superior divisors).
Rightmost column of A207375 (central divisors).
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A063538/A063539 list numbers with/without a superior prime divisor.
A070038 adds up superior divisors.
A341676 selects the unique superior prime divisor.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A059172, A116882, A116883, A341591, A341592, A341593, A341675.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A048098, A064052, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A001248, A001221, A001222, A020639, A051283.

a033677 n = head $
   dropWhile ((< n) . (^ 2)) [d | d <- [1..n], mod n d == 0]
-- Reinhard Zumkeller, Oct 20 2011

A033677 := proc(n)
    n/A033676(n) ;
end proc:

Table[Select[Divisors[n], # >= Sqrt[n] &, 1] // First, {n, 80}]  (* Jean-François Alcover, Apr 01 2011 *)

A033677(n) = {local(d); d=divisors(n); d[length(d)\2+1]} \\ Michael B. Porter, Feb 26 2010

from sympy import divisors
def A033677(n):
    d = divisors(n)
    return d[len(d)//2]  # Chai Wah Wu, Apr 05 2021

a(n) = n/A033676(n).

a(n) = A162348(2n). - Daniel Forgues, Sep 29 2014

A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Apr 26 2001

Also: Largest divisor of n which is less than sqrt(n).
If n is not a square, then a(n) = A033676(n), else a(n) is strictly smaller than A033676(n) = sqrt(n) (except for a(1) = 1).  - M. F. Hasler, Sep 20 2011
Record values occur for n = k * (k+1), for which a(n) = k. - Franklin T. Adams-Watters, May 01 2015
If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence gives the greatest strictly inferior divisor, which may differ from the lower central divisor A033676. Central divisors are listed by A207375. - Gus Wiseman, Feb 28 2021

			n = 252, D = {1, 2, 3, 4, 6, 7, 9, 12, 14, 18, 21, 28, 36, 42, 63, 84, 126, 252}, 18 divisors, the 9th is 14, so a(252) = 14.
From _Gus Wiseman_, Feb 28 2021: (Start)
The strictly inferior divisors of selected n:
n = 1  2  6  12  20  30  42  56  72  90  110  132  156  182  210  240
    -----------------------------------------------------------------
    {} 1  1  1   1   1   1   1   1   1   1    1    1    1    1    1
          2  2   2   2   2   2   2   2   2    2    2    2    2    2
             3   4   3   3   4   3   3   5    3    3    7    3    3
                     5   6   7   4   5   10   4    4    13   5    4
                                 6   6        6    6         6    5
                                 8   9        11   12        7    6
                                                             10   8
                                                             14   10
                                                                  12
                                                                  15
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000 (terms n = 2..1000 from Harry J. Smith)

Cf. A033677, A000196, A000005, A000142, A027423, A055226, A060776, A060777, A002378.
The weakly inferior version is A033676.
Positions of first appearances are A180291.
These are the row-maxima of A341674.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A070039 adds up strictly inferior divisors.
A207375 lists central divisors.
A333805 counts strictly inferior odd divisors.
A333806 counts strictly inferior prime divisors.
A341596 counts strictly inferior squarefree divisors.
A341677 counts strictly inferior prime-power divisors.
- Inferior: A063962, A066839, A069288, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A161908, A341591.
- Strictly Superior: A048098, A064052, A140271, A238535, A341642, A341673.
Cf. A000203, A001248, A006530, A020639, A112798, A161901.

with(numtheory):
a:= n-> max(select(d-> is(d=1 or dAlois P. Heinz, Jan 29 2018

Table[Part[Divisors[w], Floor[DivisorSigma[0, w]/2]], {w, 1, 256}]
Table[If[n==1,1,Max[Select[Divisors[n],#Gus Wiseman, Feb 28 2021 *)

A060775(n)=if(n>1,divisors(n)[numdiv(n)\2],1) \\ M. F. Hasler, Sep 21 2011

a(n) = max { d: d|n and d < sqrt(n) or d = 1 }, where "|" means "divides". [Corrected by M. F. Hasler, Apr 03 2019]

a(1) = 1 added (to preserve the relation a(n) | n) by Franklin T. Adams-Watters, Jan 27 2018
Edited by M. F. Hasler, Apr 03 2019
Name changed by Gus Wiseman, Feb 28 2021 (was: Lower central (median) divisor of n, with a(1) = 1.)

A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Omar E. Pol, Jun 27 2009

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

			Triangle begins:
   1....... 1;
   2....... 1;
   3....... 1;
   4..... 1,2;
   5....... 1;
   6..... 1,2;
   7....... 1;
   8..... 1,2;
   9..... 1,3;
  10..... 1,2;
  11....... 1;
  12... 1,2,3;
  13....... 1;
  14..... 1,2;
  15..... 1,3;
  16... 1,2,4;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Initial terms are A000012.
Final terms are A033676.
Row lengths are A038548 (number of inferior divisors).
Row sums are A066839 (sum of inferior divisors).
The prime terms are counted by A063962.
The odd terms are counted by A069288.
Row products are A072499.
Row LCMs are A072504.
The superior version is A161908.
The squarefree terms are counted by A333749.
The prime-power terms are counted by A333750.
The strictly superior version is A341673.
The strictly inferior version is A341674.
A001221 counts prime divisors, with sum A001414.
A000005 counts divisors, listed by A027750 with sum A000203.
A056924 count strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A217581.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000196, A001055, A001248, A006530, A020639, A050320, A068101, A161901.

a161906 n k = a161906_tabf !! (n-1) !! (k-1)
a161906_row n = a161906_tabf !! (n-1)
a161906_tabf = zipWith (\m ds -> takeWhile ((<= m) . (^ 2)) ds)
                       [1..] a027750_tabf'
-- Reinhard Zumkeller, Jun 24 2015, Mar 08 2013

div[n_] := Select[Divisors[n], # <= Sqrt[n] &]; div /@ Range[48] // Flatten (* Amiram Eldar, Nov 13 2020 *)

row(n) = select(x->(x<=sqrt(n)), divisors(n)); \\ Michel Marcus, Nov 13 2020

More terms from Sean A. Irvine, Nov 29 2010

A161908 Array read by rows in which row n lists the divisors of n that are >= sqrt(n).

Original entry on oeis.org

1, 2, 3, 2, 4, 5, 3, 6, 7, 4, 8, 3, 9, 5, 10, 11, 4, 6, 12, 13, 7, 14, 5, 15, 4, 8, 16, 17, 6, 9, 18, 19, 5, 10, 20, 7, 21, 11, 22, 23, 6, 8, 12, 24, 5, 25, 13, 26, 9, 27, 7, 14, 28, 29, 6, 10, 15, 30, 31, 8, 16, 32, 11, 33, 17, 34, 7, 35, 6, 9, 12, 18, 36, 37, 19, 38, 13, 39, 8, 10, 20, 40, 41, 7, 14, 21, 42, 43, 11, 22, 44, 9, 15, 45, 23, 46, 47, 8, 12, 16

Offset: 1

Omar E. Pol, Jun 27 2009

T(n,A038548(n)) = n. - Reinhard Zumkeller, Mar 08 2013

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by this sequence. - Gus Wiseman, Mar 08 2021

			Array begins:
1;
2;
3;
2,4;
5;
3,6;
7;
4,8;
3,9;
5,10;
11;
4,6,12;
13;
7,14;
5,15;
4,8,16;

Reinhard Zumkeller, Rows n = 1..1000 of triangle, flattened

Final terms are A000027.
Initial terms are A033677.
Row lengths are A038548 (number of superior divisors).
Row sums are A070038 (sum of superior divisors).
The inferior version is A161906.
The prime terms are counted by A341591.
The squarefree terms are counted by A341592.
The prime-power terms are counted by A341593.
The strictly superior version is A341673.
The strictly inferior version is A341674.
The odd terms are counted by A341675.
A001221 counts prime divisors, with sum A001414.
A056924 counts strictly superior (or strictly inferior divisors).
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A072500, A116882, A116883, A341592, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000005, A000203, A001055, A001248, A006530, A020639, A050320, A161901.

a161908 n k = a161908_tabf !! (n-1) !! (k-1)
a161908_row n = a161908_tabf !! (n-1)
a161908_tabf = zipWith
               (\x ds -> reverse $ map (div x) ds) [1..] a161906_tabf
-- Reinhard Zumkeller, Mar 08 2013

Table[Select[Divisors[n],#>=Sqrt[n]&],{n,100}]//Flatten (* Harvey P. Dale, Jan 01 2021 *)

More terms from Sean A. Irvine, Nov 29 2010

A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

Original entry on oeis.org

2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15, 17, 19, 20, 21, 22, 23, 25, 26, 28, 29, 31, 33, 34, 35, 37, 38, 39, 41, 42, 43, 44, 46, 47, 49, 51, 52, 53, 55, 57, 58, 59, 61, 62, 65, 66, 67, 68, 69, 71, 73, 74, 76, 77, 78, 79, 82, 83, 85, 86, 87, 88, 89, 91

Offset: 1

N. J. A. Sloane, Aug 14 2001

If we define a divisor d|n to be superior if d >= n/d, then superior divisors are counted by A038548 and listed by A161908. This sequence lists all numbers with a superior prime divisor, which is unique (A341676) when it exists. For example, 42 is in the sequence because it has a prime divisor 7 which is greater than the quotient 42/7 = 6. - Gus Wiseman, Feb 19 2021

D. H. Greene and D. E. Knuth, Mathematics for the Analysis of Algorithms; see pp. 95-98.

Robert Israel, Table of n, a(n) for n = 1..10000
Beeler, M., Gosper, R. W. and Schroeppel, R., HAKMEM, ITEM 29

Cf. A006530, A063762.
Complement of A063539. Supersequence of A001358 (semiprimes).
The strictly superior version is A064052 (complement: A048098), with associated unique prime divisor A341643.
The case of odd instead of prime divisors is A116883 (complement: A116882).
Also nonzeros of A341591 (number of superior prime divisors).
The unique superior prime divisors of the terms are A341676.
A001221 counts prime divisors, with sum A001414.
A033677 selects the smallest superior divisor.
A038548 counts superior (also inferior) divisors.
A161908 lists superior divisors.
- Inferior: A033676, A063962, A066839, A161906, A217581, A333749, A333750.
- Superior: A051283, A059172, A070038, A072500, A341592, A341593, A341675.
- Strictly Inferior: A056924, A060775, A070039, A333805, A333806, A341596, A341674, A341677.
- Strictly Superior: A140271, A238535, A341594, A341595, A341642, A341644, A341645/A341646, A341673.
Cf. A000005, A001055, A001222, A001248, A020639, A056239, A112798.

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2,seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {seq(seq(m*p, m = 1 .. min(p, N/p)),p=Primes)}:
sort(convert(S,list)); # Robert Israel, Sep 01 2015

Select[Range[2, 91], FactorInteger[#][[-1, 1]] >= Sqrt[#] &] (* Ivan Neretin, Aug 30 2015 *)

from math import isqrt
from sympy import primepi
def A063538(n):
    def f(x): return int(n+x-primepi(x//(y:=isqrt(x)))-sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Oct 05 2024

Union of A001248 and A064052. - Gus Wiseman, Feb 24 2021

A140271 Least divisor of n that is > sqrt(n), with a(1) = 1.

Original entry on oeis.org

1, 2, 3, 4, 5, 3, 7, 4, 9, 5, 11, 4, 13, 7, 5, 8, 17, 6, 19, 5, 7, 11, 23, 6, 25, 13, 9, 7, 29, 6, 31, 8, 11, 17, 7, 9, 37, 19, 13, 8, 41, 7, 43, 11, 9, 23, 47, 8, 49, 10, 17, 13, 53, 9, 11, 8, 19, 29, 59, 10, 61, 31, 9, 16, 13, 11, 67, 17, 23, 10, 71, 9, 73, 37, 15, 19, 11, 13, 79, 10, 27

Offset: 1

Leroy Quet, May 16 2008

nonn

If n is not a square, then a(n) = A033677(n).

If we define a divisor d|n to be strictly superior if d > n/d, then strictly superior divisors are counted by A056924 and listed by A341673. This sequence selects the smallest strictly superior divisor of n. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
a(n) is the smallest element in the following sets of strictly superior divisors:
   1: {1}       16: {8,16}        31: {31}
   2: {2}       17: {17}          32: {8,16,32}
   3: {3}       18: {6,9,18}      33: {11,33}
   4: {4}       19: {19}          34: {17,34}
   5: {5}       20: {5,10,20}     35: {7,35}
   6: {3,6}     21: {7,21}        36: {9,12,18,36}
   7: {7}       22: {11,22}       37: {37}
   8: {4,8}     23: {23}          38: {19,38}
   9: {9}       24: {6,8,12,24}   39: {13,39}
  10: {5,10}    25: {25}          40: {8,10,20,40}
  11: {11}      26: {13,26}       41: {41}
  12: {4,6,12}  27: {9,27}        42: {7,14,21,42}
  13: {13}      28: {7,14,28}     43: {43}
  14: {7,14}    29: {29}          44: {11,22,44}
  15: {5,15}    30: {6,10,15,30}  45: {9,15,45}
(End)

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Cf. A060775, A033676, A033677.
These divisors are counted by A056924.
These divisors add up to A238535.
These divisors that are odd are counted by A341594.
These divisors that are squarefree are counted by A341595
These divisors that are prime are counted by A341642.
These divisors are listed by A341673.
A038548 counts superior (or inferior) divisors.
A161906 lists inferior divisors.
A161908 lists superior divisors.
A207375 list central divisors.
A341674 lists strictly inferior divisors.
- Inferior: A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A051283, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A341643, A341644, A341646.
Cf. A000005, A000203, A001221, A001222, A001248, A006530, A020639, A112798.

with(numtheory):
a:= n-> min(select(d-> is(d=n or d>sqrt(n)), divisors(n))):
seq(a(n), n=1..100);  # Alois P. Heinz, Jan 29 2018

Table[Select[Divisors[n], # > Sqrt[n] &][[1]], {n, 2, 70}] (* Stefan Steinerberger, May 18 2008 *)

A140271(n)={local(d,a);d=divisors(n);a=n;for(i=1,length(d),if(d[i]>sqrt(n),a=min (d[i],a)));a} \\ Michael B. Porter, Apr 06 2010

More terms from Stefan Steinerberger, May 18 2008
a(70)-a(80) from Ray Chandler, Jun 25 2009
Franklin T. Adams-Watters, Jan 26 2018, added a(1) = 1 to preserve the relation a(n) | n.

A333805 Number of odd divisors of n that are < sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 05 2020

nonn

If we define a divisor d|n to be strictly inferior if d < n/d, then strictly inferior divisors are counted by A056924 and listed by A341674. This sequence counts strictly inferior odd divisors. - Gus Wiseman, Feb 26 2021

			The strictly inferior odd divisors of 945 are 1, 3, 5, 7, 9, 15, 21, 27, so a(945) = 8. - _Gus Wiseman_, Feb 27 2021

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

Dominated by A001227 (number of odd divisors).
Strictly inferior divisors (not just odd) are counted by A056924.
The non-strict version is A069288.
These divisors add up to A070039.
The case of prime divisors is A333806.
The strictly superior version is A341594.
The case of squarefree divisors is A341596.
The superior version is A341675.
The case of prime-power divisors is A341677.
A006530 selects the greatest prime factor.
A020639 selects the smallest prime factor.
- Odd -
A000009 counts partitions into odd parts, ranked by A066208.
A026424 lists numbers with odd Omega.
A027193 counts odd-length partitions.
A067659 counts strict partitions of odd length, ranked by A030059.
- Inferior divisors -
A033676 selects the greatest inferior divisor.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A060775 selects the greatest strictly inferior divisor.
A341674 lists strictly inferior divisors.
Cf. A001248, A051283, A063538/A063539, A063962, A116883/A116882, A300272, A333807, A333809, A340832.

Table[DivisorSum[n, 1 &, # < Sqrt[n] && OddQ[#] &], {n, 1, 90}]
nmax = 90; CoefficientList[Series[Sum[x^(2 k (2 k - 1))/(1 - x^(2 k - 1)), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

A333805(n) = sumdiv(n,d,(d%2)&&((d*d)Antti Karttunen, Nov 02 2022

G.f.: Sum_{k>=1} x^(2*k*(2*k - 1)) / (1 - x^(2*k - 1)).

Data section extended up to a(105) by Antti Karttunen, Nov 02 2022

A341674 Irregular triangle read by rows giving the strictly inferior divisors of n.

Original entry on oeis.org

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

Offset: 1

Gus Wiseman, Feb 23 2021

We define a divisor d|n to be strictly inferior if d < n/d. The number of strictly inferior divisors of n is A056924(n).

			Triangle begins:
     1: {}        16: 1,2        31: 1
     2: 1         17: 1          32: 1,2,4
     3: 1         18: 1,2,3      33: 1,3
     4: 1         19: 1          34: 1,2
     5: 1         20: 1,2,4      35: 1,5
     6: 1,2       21: 1,3        36: 1,2,3,4
     7: 1         22: 1,2        37: 1
     8: 1,2       23: 1          38: 1,2
     9: 1         24: 1,2,3,4    39: 1,3
    10: 1,2       25: 1          40: 1,2,4,5
    11: 1         26: 1,2        41: 1
    12: 1,2,3     27: 1,3        42: 1,2,3,6
    13: 1         28: 1,2,4      43: 1
    14: 1,2       29: 1          44: 1,2,4
    15: 1,3       30: 1,2,3,5    45: 1,3,5

Initial terms are A000012.
Row lengths are A056924 (number of strictly inferior divisors).
Final terms are A060775.
Row sums are A070039 (sum of strictly inferior divisors).
The weakly inferior version is A161906.
The weakly superior version is A161908.
The odd terms are counted by A333805.
The prime terms are counted by A333806.
The squarefree terms are counted by A341596.
The strictly superior version is A341673.
The prime-power terms are counted by A341677.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A207375 lists central divisors.
- Inferior: A033676, A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Superior: A048098, A064052, A140271, A238535, A341594, A341595, A341642, A341643, A341644, A341645, A341646.
Cf. A000005, A000203, A001055, A001248, A006530, A020639, A050320.

Mathematica
```
Table[Select[Divisors[n],#
				
```

A341673 Irregular triangle read by rows giving the strictly superior divisors of n.

Original entry on oeis.org

2, 3, 4, 5, 3, 6, 7, 4, 8, 9, 5, 10, 11, 4, 6, 12, 13, 7, 14, 5, 15, 8, 16, 17, 6, 9, 18, 19, 5, 10, 20, 7, 21, 11, 22, 23, 6, 8, 12, 24, 25, 13, 26, 9, 27, 7, 14, 28, 29, 6, 10, 15, 30, 31, 8, 16, 32, 11, 33, 17, 34, 7, 35, 9, 12, 18, 36, 37, 19, 38, 13, 39

Offset: 1

Gus Wiseman, Feb 22 2021

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

			Row n = 18 lists the strictly superior divisors of 18, which are 6, 9, 18.
Triangle begins:
   1: {}
   2: 2
   3: 3
   4: 4
   5: 5
   6: 3,6
   7: 7
   8: 4,8
   9: 9
  10: 5,10
  11: 11
  12: 4,6,12
  13: 13
  14: 7,14
  15: 5,15
  16: 8,16
  17: 17
  18: 6,9,18
  19: 19
  20: 5,10,20

Final terms in each row (except n = 1) are A000027.
Row lengths are A056924 (number of strictly superior divisors).
Initial terms in each row are A140271.
The weakly inferior version is A161906.
The weakly superior version is A161908.
Row sums are A238535.
The odd terms in each row are counted by A341594.
The squarefree terms in each row are counted by A341595.
The prime terms in each row are counted by A341642.
The strictly inferior version is A341674.
A001221 counts prime divisors, with sum A001414.
A038548 counts superior (or inferior) divisors.
A207375 list central divisors.
- Inferior: A033676, A063962, A066839, A069288, A217581, A333749, A333750.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A072500, A116882, A116883, A341591, A341592, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341596, A341677.
- Strictly Superior: A048098, A064052, A341643, A341644, A341646.
Cf. A000005, A000203, A001222, A001248, A006530, A020639.

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

A333749 Number of squarefree divisors of n that are <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Ilya Gutkovskiy, Apr 03 2020

nonn

If we define a divisor d|n to be inferior if d <= n/d, then inferior divisors are counted by A038548 and listed by A161906. This sequence counts inferior squarefree divisors. - Gus Wiseman, Feb 27 2021

			   n   inferior squarefree divisors of n
  ---  ---------------------------------
   33  1,  3
   56  1,  2,  7
  429  1,  3, 11, 13
   90  1,  2,  3,  5,  6
  490  1,  2,  5,  7, 10, 14
  480  1,  2,  3,  5,  6, 10, 15

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A034444, A069291, A333748, A333752.
Positions of 1's are A008578.
The case of equality is the indicator function of A062503.
The version for prime instead of squarefree divisors is A063962.
The version for odd instead of squarefree divisors is A069288.
The version for prime-power instead of squarefree divisors is A333750.
The superior version is A341592.
The strictly superior version is A341595.
The strictly inferior version is A341596.
A005117 lists squarefree numbers.
A038548 counts superior (or inferior) divisors.
A056924 counts strictly superior (or strictly inferior) divisors.
A161906 lists inferior divisors.
A161908 lists superior divisors.
A207375 list central divisors.
- Inferior: A033676, A066839, A217581.
- Superior: A033677, A051283, A059172, A063538, A063539, A070038, A116882, A116883, A341593, A341675, A341676.
- Strictly Inferior: A060775, A070039, A333805, A333806, A341674, A341677.
- Strictly Superior: A048098, A064052 A140271, A238535, A341591, A341594, A341642, A341643, A341644, A341645/A341646, A341673.
Cf. A000005, A000203, A001221, A001222, A001248, A006530, A020639.

N:= 200: # for a(1)..a(N)
g:= add(x^(k^2)/(1-x^k), k = select(numtheory:-issqrfree,[$1..floor(sqrt(N))])):
S:= series(g,x,N+1):
seq(coeff(S,x,j),j=1..N); # Robert Israel, Apr 05 2020

Table[DivisorSum[n, 1 &, # <= Sqrt[n] && SquareFreeQ[#] &], {n, 1, 100}]
nmax = 100; CoefficientList[Series[Sum[MoebiusMu[k]^2 x^(k^2)/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x] // Rest

a(n) = sumdiv(n, d, (d^2<=n) && issquarefree(d)); \\ Michel Marcus, Apr 03 2020

G.f.: Sum_{k>=1} mu(k)^2 * x^(k^2) / (1 - x^k).

cp's OEIS Frontend

A033677 Smallest divisor of n >= sqrt(n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Formula

A060775 The greatest divisor d|n such that d < n/d, with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

Extensions

A161906 Triangle read by rows in which row n lists the divisors of n that are <= sqrt(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

PARI

Extensions

A161908 Array read by rows in which row n lists the divisors of n that are >= sqrt(n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Mathematica

Extensions

A063538 Numbers n that are not sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) >= sqrt(n).

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Maple

Mathematica

Python

Formula

A140271 Least divisor of n that is > sqrt(n), with a(1) = 1.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Maple

Mathematica