A066839 -id:A066839 - cp's OEIS frontend

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

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

A063962 Number of distinct prime divisors of n that are <= sqrt(n).

Original entry on oeis.org

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

Offset: 1

Reinhard Zumkeller, Sep 04 2001

nonn

For all primes p: a(p) = 0 (not marked) and for k > 1 a(p^k) = 1.
a(1) = 0 and for n > 0 a(n) is the number of marks when applying the sieve of Eratosthenes where a stage for prime p starts at p^2.
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 prime divisors. - Gus Wiseman, Feb 25 2021

			a(33) = a(3*11) = 1, as 3^2 = 9 < 33 and 11^2 = 121 > 33.
From _Gus Wiseman_, Feb 25 2021: (Start)
The a(n) inferior prime divisors (columns) for selected n:
n =  3  8  24  3660  390  3570 87780
   ---------------------------------
    {}  2   2     2    2     2     2
            3     3    3     3     3
                  5    5     5     5
                      13     7     7
                            17    11
                                  19
(End)

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A055399, A001221.
Cf. A027748, A063962.
Zeros are at indices A008578.
The divisors are listed by A161906 and add up to A097974.
Dominates A333806 (the strictly inferior version).
The superior version is A341591.
The strictly superior version is A341642.
A001221 counts prime divisors, with sum A001414.
A033677 selects the smallest superior divisor.
A038548 counts inferior divisors.
A063538/A063539 have/lack a superior prime divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
A217581 selects the greatest inferior prime divisor.
A341676 lists the unique superior prime divisors.
- Inferior: A033676, A066839, A069288, A072499, A333749, A333750.
- Superior: A051283, A059172, A070038, A072500, A116883, A341592, A341675.
- Strictly Inferior: A056924, A060775, A070039, A333805, A341596, A341674.
- Strictly Superior: A056924, A140271, A238535, A341594, A341595, A341673.
Cf. A000005, A001248, A005117, A006530, A020639, A048098, A064052, A341643.

a063962 n = length [p | p <- a027748_row n, p ^ 2 <= n]
-- Reinhard Zumkeller, Apr 05 2012

with(numtheory): a:=proc(n) local c,F,f,i: c:=0: F:=factorset(n): f:=nops(F): for i from 1 to f do if F[i]^2<=n then c:=c+1 else c:=c: fi od: c; end: seq(a(n),n=1..105); # Emeric Deutsch

Join[{0},Table[Count[Transpose[FactorInteger[n]][[1]],?(#<=Sqrt[n]&)],{n,2,110}]] (* _Harvey P. Dale, Mar 26 2015 *)

{ for (n=1, 1000, f=factor(n)~; a=0; for (i=1, length(f), if (f[1, i]^2<=n, a++, break)); write("b063962.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 04 2009

G.f.: Sum_{k>=1} x^(prime(k)^2) / (1 - x^prime(k)). - Ilya Gutkovskiy, Apr 04 2020

a(A002110(n)) = n for n > 2. - Gus Wiseman, Feb 25 2021

Revised definition from Emeric Deutsch, Jan 31 2006

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.

A238535 Sum of divisors d of n where d > sqrt(n).

Original entry on oeis.org

0, 2, 3, 4, 5, 9, 7, 12, 9, 15, 11, 22, 13, 21, 20, 24, 17, 33, 19, 35, 28, 33, 23, 50, 25, 39, 36, 49, 29, 61, 31, 56, 44, 51, 42, 75, 37, 57, 52, 78, 41, 84, 43, 77, 69, 69, 47, 108, 49, 85, 68, 91, 53, 108, 66, 106, 76, 87, 59, 147, 61, 93, 93, 112, 78, 132

Offset: 1

Michel Lagneau, Feb 28 2014

nonn

Properties of the sequence:
a(n) = n if n is prime because sigma(n) = n+1 and A066839(n) = 1;
a(p^2) = p^2 if p is prime because sigma(p^2) = p^2+p+1 and A066839(p^2)= p+1 => A000203(p^2) - A066839(p^2)= p^2;
a(m) = 2*m if m = A182147(n) = 42, 54, 66, 78, 102, 114,... (numbers n equal to the sum of its proper divisors greater than square root of n).

			a(8) = 12 because A000203(8)= 15 and A066839(8) = 3 => 15 - 8 = 12.

Michel Lagneau, Table of n, a(n) for n = 1..10000

Cf. A000203, A066839, A182147, A238502.

lst={}; f[n_]:=DivisorSigma[1,n]-Plus@@Select[Divisors@n,#<=Sqrt@n&];Do[If[IntegerQ[f[n]],AppendTo[lst, f[n]]],{n,1,200}];lst

a(n) = sumdiv(n, d, d*(d>sqrt(n))); \\ Michel Marcus, Feb 28 2014

def a(n):
    return sum([d for d in Integer(n).divisors() if d>sqrt(n)]) # Ralf Stephan, Mar 08 2014

a(n) = A000203(n) - A066839(n).

Better name from Ralf Stephan, Mar 08 2014

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],#
				
```

A037213 Expansion of Sum_{n>=0} n*q^(n^2).

Original entry on oeis.org

0, 1, 0, 0, 2, 0, 0, 0, 0, 3, 0, 0, 0, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, 0, 5, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 7, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 8, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane

Multiplicative with a(p^(2e)) = p^e, a(p^(2e+1)) = 0. - Mitch Harris, Jun 09 2005
a(n) is the square root of n if n is square, zero otherwise. - Carl R. White, May 23 2009
Möbius transform of A069290(n). - Wesley Ivan Hurt, Jul 09 2025

Reinhard Zumkeller, Table of n, a(n) for n = 0..10000

Cf. A000196, A008683 (mu), A010052, A066839, A069290, A070039.

a037213 n = if n == r ^ 2 then r else 0  where r = a000196 n
a037213_list = zipWith (*) a010052_list a000196_list
-- Reinhard Zumkeller, Nov 09 2012

Table[If[IntegerQ[n^(1/2)], n^(1/2), 0], {n, 0, 100}] (* Geoffrey Critzer, Feb 21 2015 *)

a(n) = if (issquare(n), sqrtint(n), 0); \\ Michel Marcus, Aug 21 2025

Dirichlet generating function: zeta(2*s-1). - Franklin T. Adams-Watters, Sep 11 2005
a(n) = sqrt(n) * floor( cos^2(Pi * sqrt(n)) ). - Carl R. White, May 23 2009
a(n) = A000196(n) * A010052(n). - Reinhard Zumkeller, Jan 27 2010
Sum_{k=1..n} a(k) ~ n/2. - Vaclav Kotesovec, Aug 19 2021
a(n) = A066839(n) - A070039(n). - Ridouane Oudra, Jun 24 2025
a(n) = Sum_{d|n} A069290(d) * mu(n/d). - Wesley Ivan Hurt, Jul 09 2025

A217581 Largest prime divisor of n <= sqrt(n), 1 if n is prime or 1.

Original entry on oeis.org

1, 1, 1, 2, 1, 2, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 1, 2, 3, 2, 1, 3, 5, 2, 3, 2, 1, 5, 1, 2, 3, 2, 5, 3, 1, 2, 3, 5, 1, 3, 1, 2, 5, 2, 1, 3, 7, 5, 3, 2, 1, 3, 5, 7, 3, 2, 1, 5, 1, 2, 7, 2, 5, 3, 1, 2, 3, 7, 1, 3, 1, 2, 5, 2, 7, 3, 1, 5, 3, 2, 1, 7, 5, 2, 3

Offset: 1

Peter Luschny, Mar 21 2013

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 selects the greatest inferior prime divisor of n. - Gus Wiseman, Apr 06 2021

			From _Gus Wiseman_, Apr 06 2021: (Start)
The sequence selects the greatest element (or 1 if empty) of each of the following sets of strictly superior divisors:
   1:{}     16:{2}      31:{}     46:{2}
   2:{}     17:{}       32:{2}    47:{}
   3:{}     18:{2,3}    33:{3}    48:{2,3}
   4:{2}    19:{}       34:{2}    49:{7}
   5:{}     20:{2}      35:{5}    50:{2,5}
   6:{2}    21:{3}      36:{2,3}  51:{3}
   7:{}     22:{2}      37:{}     52:{2}
   8:{2}    23:{}       38:{2}    53:{}
   9:{3}    24:{2,3}    39:{3}    54:{2,3}
  10:{2}    25:{5}      40:{2,5}  55:{5}
  11:{}     26:{2}      41:{}     56:{2,7}
  12:{2,3}  27:{3}      42:{2,3}  57:{3}
  13:{}     28:{2}      43:{}     58:{2}
  14:{2}    29:{}       44:{2}    59:{}
  15:{3}    30:{2,3,5}  45:{3,5}  60:{2,3,5}
(End)

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

Cf. A033676.
Positions of first appearances are 1 and A001248.
These divisors are counted by A063962.
These divisors add up to A097974.
The smallest prime factor of the same type is A107286.
A strictly superior version is A341643.
A superior version is A341676.
A038548 counts superior (or inferior) divisors.
A048098 lists numbers without a strictly superior prime divisor.
A056924 counts strictly superior (or strictly inferior) divisors.
A063538/A063539 have/lack a superior prime divisor.
A140271 selects the smallest strictly superior divisor.
A161906 lists inferior divisors.
A207375 lists central divisors.
A341591 counts superior prime divisors.
A341642 counts strictly superior prime divisors.
A341673 lists strictly superior divisors.
- Inferior: A066839, A069288, A333749, A333750.
- Superior: A033677, A051283, A059172, A070038, A116882, A116883, A161908, A341592, A341593, A341675.
- Strictly Inferior: A060775, A333805, A333806, A341596, A341674.
- Strictly Superior: A238535, A341594, A341595, A341644, A341645, A341646.
Cf. A000005, A001055, A001221, A001222, A001248, A001414, A006530, A020639, A064052.

A217581 := n -> `if`(isprime(n) or n=1, 1, max(op(select(i->i^2<=n, numtheory[factorset](n)))));

Table[If[n == 1 || PrimeQ[n], 1, Select[Transpose[FactorInteger[n]][[1]], # <= Sqrt[n] &][[-1]]], {n, 100}] (* T. D. Noe, Mar 25 2013 *)

a(n) = {my(m=1); foreach(factor(n)[,1], d, if(d^2 <= n, m=max(m,d))); m} \\ Andrew Howroyd, Oct 11 2023

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).

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

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A063962 Number of distinct prime divisors of n that are <= sqrt(n).

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

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

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A238535 Sum of divisors d of n where d > sqrt(n).

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

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A037213 Expansion of Sum_{n>=0} n*q^(n^2).

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