A070039 -id:A070039 - cp's OEIS frontend

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

2, 3, 4, 186, 318, 434, 473, 582, 730, 978, 1024, 1035, 1245, 1357, 1397, 1506, 1661, 1902, 2085, 2116, 2224, 2329, 2453, 2505, 2506, 2770, 2954, 3144, 3345, 3377, 3624, 3641, 3765, 3790, 3882, 4037, 4172, 4438, 4898, 4938, 4975, 5221, 6126, 6285, 6312, 6356

Offset: 1

Amiram Eldar, Sep 03 2019

nonn

			186 is in the sequence since A070039(186) = A070039(187) = 12.

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

Cf. A002961, A064115, A064125, A070039, A164522, A164557, A293183, A306985, A325175.

s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; Do[s2 = s[n]; If[s1 == s2, AppendTo[seq, n - 1]]; s1 = s2, {n, 2, 6500}]; seq

A325175 Numbers k such that s(k) = s(k+1) = s(k+2) = s(k+3) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 1190499411, 7896840789, 9190223334, 10546199943, 13239147654, 16488284454, 19818736503, 19927041063, 20401914903, 20551875063, 24219563814, 25442242503, 27391724583, 34539825543, 35714513463, 37014168183, 44120613543, 44130940614, 44514172134, 48647900639

Offset: 1

Giovanni Resta, Sep 05 2019

nonn

Five consecutive coincident values of A070039 are also possible. One such occurrence starts a 211301122224870.

			2 is in the sequence since A070039(2) = A070039(3) = A070039(4) = A070039(5) = 1.

Cf. A070039, A324295, A324310.

A324310 Numbers k such that s(k) = s(k+1) = s(k+2) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

Original entry on oeis.org

2, 3, 2505, 64486, 113413, 205365, 423414, 496156, 635053, 664033, 881565, 1011793, 1190685, 1306605, 1442136, 1923655, 1947766, 2286913, 2324422, 2465805, 3030733, 3291553, 3335205, 4100086, 4547353, 4648965, 4987065, 5025705, 5034904, 5069113, 5827485, 5909413

Offset: 1

Amiram Eldar, Sep 03 2019

nonn

Are there 4 consecutive numbers with the same value of A070039, apart from 2, 3, 4, 5?

The next 4 consecutive numbers with the same value of A070039 start at 1190499411. - Giovanni Resta, Sep 05 2019

			2 is in the sequence since A070039(2) = A070039(3) = A070039(4) = 1.

Cf. A070039, A324295, A325175.

s[n_] := DivisorSum[n, # &, # < Sqrt[n] &]; seq={}; s1 = 0; s2=0; Do[s3 = s[n]; If[s1 == s2 && s2 == s3, AppendTo[seq, n - 2]]; s1 = s2; s2 = s3, {n, 2, 10^5}]; seq

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

Original entry on oeis.org

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

A056924 Number of divisors of n that are smaller than sqrt(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, 3, 2, 2, 1, 4, 1, 2, 2, 3, 1, 4, 1, 3, 2, 2, 2, 4, 1, 2, 2, 4, 1, 4, 1, 3, 3, 2, 1, 5, 1, 3, 2, 3, 1, 4, 2, 4, 2, 2, 1, 6, 1, 2, 3, 3, 2, 4, 1, 3, 2, 4, 1, 6, 1, 2, 3, 3, 2, 4, 1, 5, 2, 2, 1, 6, 2, 2, 2, 4, 1, 6, 2, 3, 2, 2, 2, 6, 1, 3, 3, 4, 1, 4, 1, 4, 4

Offset: 1

Henry Bottomley, Jul 12 2000

nonn

Number of powers of n in product of factors of n if n>1.
Also, the number of solutions to the Pell equation x^2 - y^2 = 4n. - Ralf Stephan, Sep 20 2013
If n is a prime or the square of a prime, then a(n)=1.
Number of positive integer solutions to the equation x^2 + k*x - n = 0, for all k in 1 <= k <= n. - Wesley Ivan Hurt, Dec 27 2020
Number of pairs of distinct divisors (d,n/d) of n, with dWesley Ivan Hurt, Nov 09 2023

			a(16)=2 since the divisors of 16 are 1,2,4,8,16 of which 2 are less than sqrt(16) = 4.
From _Labos Elemer_, Apr 19 2002: (Start)
n=96: a(96) = Card[{1,2,3,4,6,8}] = 6 = Card[{12,16,24,32,48,96}];
n=225: a(225) = Card[{1,3,5,9}] = Card[{15,25,45,7,225}]-1. (End)

T. D. Noe, Table of n, a(n) for n=1..10000
Cristina Ballantine and Mircea Merca, New convolutions for the number of divisors, Journal of Number Theory, Vol. 170 (2016), pp. 17-34.
S.-H. Cha, E. G. DuCasse, and L. V. Quintas, Graph invariants based on the divides relation and ordered by prime signatures, arXiv:1405.5283 [math.NT], 2014, eq. (2.29).

Cf. A038548, A000203, A000005, A070038, A070039.

Cf. A227068 (records).

a056924 = (`div` 2) . a000005  -- Reinhard Zumkeller, Jul 12 2013

with(numtheory); A056924 := n->floor(tau(n)/2); seq(A056924(k),k=1..100); # Wesley Ivan Hurt, Jun 14 2013

di[x_] := Divisors[x] lds[x_] := Ceiling[DivisorSigma[0, x]/2] rd[x_] := Reverse[Divisors[x]] td[x_] := Table[Part[rd[x], w], {w, 1, lds[x]}] sud[x_] := Apply[Plus, td[x]] Table[DivisorSigma[0, w]-lds[w], {w, 1, 128}] (* Labos Elemer, Apr 19 2002 *)
Table[Length[Select[Divisors[n], # < Sqrt[n] &]], {n, 100}] (* T. D. Noe, Jul 11 2013 *)
a[n_] := Floor[DivisorSigma[0, n]/2]; Array[a, 100] (* Amiram Eldar, Jun 26 2022 *)

a(n)=if(n<1, 0, numdiv(n)\2) /* Michael Somos, Mar 18 2006 */

from sympy import divisor_count
def A056924(n): return divisor_count(n)//2 # Chai Wah Wu, Jun 25 2022

For n>1, a(n) = floor[log(A007955(n))/log(n)] = log(A056925(n))/log(n) = floor[d(n)/2] = floor[A000005(n)/2] = ( A000005(n)-A010052(n) )/2.
a(n) = A000005(n) - A038548(n). - Labos Elemer, Apr 19 2002
G.f.: Sum_{k>0} x^(k^2+k)/(1-x^k). - Michael Somos, Mar 18 2006
a(n) = (1/2) * Sum_{d|n} (1 - [d = n/d]), where [ ] is the Iverson bracket. - Wesley Ivan Hurt, Jan 28 2021

Edited by Michael Somos, Mar 18 2006

A066839 a(n) = sum of positive divisors k of n with k <= sqrt(n).

Original entry on oeis.org

1, 1, 1, 3, 1, 3, 1, 3, 4, 3, 1, 6, 1, 3, 4, 7, 1, 6, 1, 7, 4, 3, 1, 10, 6, 3, 4, 7, 1, 11, 1, 7, 4, 3, 6, 16, 1, 3, 4, 12, 1, 12, 1, 7, 9, 3, 1, 16, 8, 8, 4, 7, 1, 12, 6, 14, 4, 3, 1, 21, 1, 3, 11, 15, 6, 12, 1, 7, 4, 15, 1, 24, 1, 3, 9, 7, 8, 12, 1, 20, 13, 3, 1, 23, 6, 3, 4, 15, 1, 26, 8, 7, 4, 3

Offset: 1

Leroy Quet, Jan 20 2002

nonn

Row sums of the table in A161906. - Reinhard Zumkeller, Mar 08 2013
Conjecture: a(n) is the total number of parts in all partitions of n into consecutive parts that differ by 2. - Omar E. Pol, May 03 2020. This conjecture is true (the g.f. for these partitions agrees with the g.f. given below by Michael Somos). - N. J. A. Sloane, Dec 02 2020
Column 2 of A334466. - Omar E. Pol, Dec 03 2020

			a(9) = 4 = 1 + 3 because 1 and 3 are the positive divisors of 9 that are <= sqrt(9).
a(20) = 7: the divisors of 20 are 1, 2, 4, 5, 10 and 20. a(20) = 1 + 2 + 4 = 7.

Vaclav Kotesovec, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)
Douglas E. Iannucci, On sums of the small divisors of a natural number, arXiv:1910.11835 [math.NT], 2019.

Cf. A000203, A070038, A038548, A072499, A334466.

a066839 = sum . a161906_row  -- Reinhard Zumkeller, Mar 08 2013

with(numtheory):for n from 1 to 200 do c[n] := 0:d := divisors(n):for i from 1 to nops(d) do if d[i]<=n^.5+10^(-10) then c[n] := c[n]+d[i]:fi:od:od:seq(c[i],i=1..200);
# alternative
seq(add(d, d in select(x->x^2<=n, numtheory[divisors](n))), n=1..100); # Ridouane Oudra, Jun 24 2025

f[n_] := Plus @@ Select[ Divisors@n, # <= Sqrt@n &]; Array[f, 94] (* Robert G. Wilson v, Mar 04 2010 *)
Table[Sum[If[n > k*(k-1), k, 0], {k, Divisors[n]}], {n, 1, 100}] (* Vaclav Kotesovec, Oct 22 2024 *)

a(n)=sumdiv(n,d, (d^2<=n)*d) /* Michael Somos, Nov 19 2005 */

{ for (n=1, 1000, d=divisors(n); s=sum(k=1, ceil(length(d)/2), d[k]); write("b066839.txt", n, " ", s) ) } \\ Harry J. Smith, Mar 31 2010

from itertools import takewhile
from sympy import divisors
def A066839(n): return sum(takewhile(lambda x:x**2<=n,divisors(n))) # Chai Wah Wu, Dec 19 2023

[sum(k for k in divisors(n) if k^2<=n) for n in (1..94)] # Giuseppe Coppoletta, Jan 21 2015

G.f.: Sum_{k>0} k*x^(k^2)/(1-x^k). - Michael Somos, Nov 19 2005
a(n) = Sum_{i=1..floor(sqrt(n))} (-(n mod i) + (n-1) mod i + 1). - José de Jesús Camacho Medina, Feb 21 2021
a(p^(2k+1)) = a(p^(2k)) = (p^(k+1)-1)/(p-1) = A000203(p^k) for k>=0 and p prime. - Chai Wah Wu, Dec 23 2023
Sum_{k=1..n} a(k) ~ 2 * n^(3/2) / 3 [Iannucci, 2019]. - Vaclav Kotesovec, Oct 23 2024
a(n) = A070039(n) + A037213(n). - Ridouane Oudra, Jun 24 2025

More terms from Larry Reeves (larryr(AT)acm.org), Apr 12 2002

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.)

A063539 Numbers n that are sqrt(n-1)-smooth: largest prime factor of n (=A006530(n)) < sqrt(n).

Original entry on oeis.org

1, 8, 12, 16, 18, 24, 27, 30, 32, 36, 40, 45, 48, 50, 54, 56, 60, 63, 64, 70, 72, 75, 80, 81, 84, 90, 96, 98, 100, 105, 108, 112, 120, 125, 126, 128, 132, 135, 140, 144, 147, 150, 154, 160, 162, 165, 168, 175, 176, 180, 182, 189, 192, 195, 196

Offset: 1

N. J. A. Sloane, Aug 14 2001

nonn

Sometimes (Weisstein) called the "usual numbers" as opposed to what Greene and Knuth define as "unusual numbers" (A063538), which turn out to not be so unusual after all (Greene and Knuth 1990, Finch 2001). - Jonathan Vos Post, Sep 11 2010
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 numbers without a superior prime divisor, which is unique (A341676) when it exists. For example, the set of superior prime divisors of each n starts: {},{2},{3},{2},{5},{3},{7},{},{3},{5},{11},{},{13},{7}. The positions of empty sets give the sequence. - Gus Wiseman, Feb 24 2021
As Jonathan Vos Post's comment suggests, the sqrt(n-1)-smooth numbers are asymptotically less dense than their "unusual" complement. This is part of a larger picture of "typical" relative sizes of a number's prime factors: see, for example, the medians of the n-th smallest prime factors of the positive integers in A281889. - Peter Munn, Mar 03 2021

			a(100) = 360; a(1000) = 3744; a(10000) = 37665; a(100000)=375084;
a(10^6) = 3697669; a(10^7) = 36519633; a(10^8) = 360856296;
a(10^9) = 3571942311; a(10^10) = 35410325861; a(10^11) = 351498917129. - _Giovanni Resta_, Apr 12 2020

Greene, D. H. and Knuth, D. E., Mathematics for the Analysis of Algorithms, 3rd ed. Boston, MA: Birkhäuser, pp. 95-98, 1990.

N. J. A. Sloane, Table of a(n) for n = 1..10622 [Extending and correcting earlier b-files from T. D. Noe and Marius A. Burtea]
M. Beeler, R. W. Gosper and R. Schroeppel, HAKMEM, ITEM 29
Steven Finch, "RE: Unusual Numbers.", Aug 27 2001.
Hugo Pfoertner, Illustration of deviation from linear fit 3.7642*n, (2020).
Hugo Pfoertner, Illustration of relative error of asymptotic approximation, (2020).
Project Euler, Problem 668: Square root smooth numbers
V. Ramaswami, On the number of positive integers less than x and free of prime divisors greater than x^c, Bull. Amer. Math. Soc. 55 (1949), 1122-1127.
Eric W. Weisstein, Rough Number. [From _Jonathan Vos Post_, Sep 11 2010]
Wikipedia, Dickman function

Set difference of A048098 and A001248.
Complement of A063538.
Cf. A006530.
The following are all different versions of sqrt(n)-smooth numbers: A048098, A063539, A064775, A295084, A333535, A333536.
Positions of zeros in A341591.
A001221 counts prime divisors, with sum A001414.
A001222 counts prime-power divisors.
A033677 selects the smallest superior divisor.
A038548 counts superior (or inferior) divisors.
A051283 lists numbers without a superior prime-power divisor.
A056924 counts strictly superior (or strictly inferior) divisors.
A059172 lists numbers without a superior squarefree divisor.
A063962 counts inferior prime divisors.
A116882/A116883 list numbers with/without a superior odd divisor.
A161908 lists superior divisors.
A207375 lists central divisors.
A217581 selects the greatest inferior prime divisor.
A341642 counts strictly superior prime divisors.
A341676 gives unique superior prime divisors, with strict case A341643.
- Inferior: A033676, A066839, A069288, A161906, A333749, A333750.
- Superior: A070038, A072500, A341592, A341593, A341675.
- Strictly inferior: A060775, A070039, A333805, A333806, A341596, A341674.
- Strictly superior: A064052, A140271, A238535, A341594, A341595, A341644, A341645, A341646, A341673.
Cf. A000005, A000203, A020639, A112798, A281889.

[1] cat [m:m in [2..200]| Max(PrimeFactors(m)) lt Sqrt(m) ]; // Marius A. Burtea, May 08 2019

N:= 1000: # to get all terms <= N
Primes:= select(isprime, [2, seq(2*i+1, i=1..floor((N-1)/2))]):
S:= {$1..N} minus {seq(seq(m*p, m = 1 .. min(p, N/p)), p=Primes)}:
sort(convert(S, list)); # Robert Israel, Sep 02 2015

Prepend[Select[Range[192], FactorInteger[#][[-1, 1]] < Sqrt[#] &], 1] (* Ivan Neretin, Sep 02 2015 *)

from math import isqrt
from sympy import primepi
def A063539(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+primepi(x//(y:=isqrt(x)))+sum(primepi(x//i)-primepi(i) for i in range(1,y)))
    return bisection(f,n,n) # Chai Wah Wu, Oct 05 2024

From Hugo Pfoertner, Apr 02 - Apr 12 2020: (Start)
For small n (e.g. n < 10000) a(n) can apparently be approximated by 3.7642*n.
Asymptotically, the number of sqrt(n)-smooth numbers < x is known to be (1-log(2))*x + O(x/log(x)), see Ramaswami (1949).
n = (1-log(2))*a(n) - 0.59436*a(n)/log(a(n)) is a fitted approximation. (End)
However, it is known that this fit only leads to an increase of accuracy in the range up to a(10^11). The improvement in accuracy suggested by the plot of the relative error for even larger n does not occur. For larger n the behavior of the error term O(x/log(x)) is not known. - Hugo Pfoertner, Nov 12 2023

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

A070038 a(n) = sum of divisors of n that are at least sqrt(n).

Original entry on oeis.org

1, 2, 3, 6, 5, 9, 7, 12, 12, 15, 11, 22, 13, 21, 20, 28, 17, 33, 19, 35, 28, 33, 23, 50, 30, 39, 36, 49, 29, 61, 31, 56, 44, 51, 42, 81, 37, 57, 52, 78, 41, 84, 43, 77, 69, 69, 47, 108, 56, 85, 68, 91, 53, 108, 66, 106, 76, 87, 59, 147, 61, 93, 93, 120, 78, 132, 67, 119, 92

Offset: 1

Labos Elemer, Apr 19 2002

nonn

a(n) = n iff n is not a composite number.

Sum of a subset of all divisors of n, not including complementary divisors of any term.

			a(20) = 35: the divisors of 20 are 1,2,4,5,10 and 20. a(20) = 5 + 10 + 20 = 35.
a(96) = 228 = 96 + 48 + 32 + 24 + 16 + 12 (sum of an even number of divisors);
a(225) = 385 = 225 + 75 + 45 + 25 + 15 (sum of an odd number of divisors).

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Cf. A038548, A000203, A000005, A070039, A056924, A072500, A066839.

with(numtheory):for n from 1 to 200 do c[n] := 0:d := divisors(n):for i from 1 to nops(d) do if d[i]>=n^.5 then c[n] := c[n]+d[i]:fi:od:od:seq(c[i],i=1..200);

Table[Plus @@ Select[Divisors[n], # >= Sqrt[n] &], {n, 1, 70}]

a(n) = sumdiv(n, d, d*(d^2>=n)); \\ Michel Marcus, Jan 22 2015

[sum(k for k in divisors(n) if k^2>=n) for n in range (1,70)] # Giuseppe Coppoletta, Jan 21 2015

cp's OEIS Frontend

A324295 Numbers k such that s(k) = s(k+1) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

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A325175 Numbers k such that s(k) = s(k+1) = s(k+2) = s(k+3) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

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A324310 Numbers k such that s(k) = s(k+1) = s(k+2) where s(k) is the sum of divisors of k that are smaller than sqrt(k) (A070039).

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

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A056924 Number of divisors of n that are smaller than sqrt(n).

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A066839 a(n) = sum of positive divisors k of n with k <= sqrt(n).

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

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