A161901 -id:A161901 - cp's OEIS frontend

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

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

A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 5, 2, 3, 1, 7, 2, 4, 3, 2, 5, 1, 11, 3, 4, 1, 13, 2, 7, 3, 5, 4, 1, 17, 3, 6, 1, 19, 4, 5, 3, 7, 2, 11, 1, 23, 4, 6, 5, 2, 13, 3, 9, 4, 7, 1, 29, 5, 6, 1, 31, 4, 8, 3, 11, 2, 17, 5, 7, 6, 1, 37, 2, 19, 3, 13, 5, 8, 1, 41, 6, 7, 1, 43

Offset: 1

Omar E. Pol, Feb 23 2012

If n is a square then row n lists only the square root of n because the squares (A000290) have only one central divisor.
If n is not a square then row n lists the pair (j, k) of divisors of n, nearest to the square root of n, such that j*k = n.
Conjecture 1: It appears that the n-th record in this sequence is the last member of row A008578(n).
Column 1 gives A033676. Right border gives A033677. - Omar E. Pol, Feb 26 2019
The conjecture 1 follows from Bertrand's Postulate. - Charles R Greathouse IV, Feb 11 2022
Row products give A097448. - Omar E. Pol, Feb 17 2022

			Array begins:
  1;
  1,  2;
  1,  3;
  2;
  1,  5;
  2,  3;
  1,  7;
  2,  4;
  3;
  2,  5;
  1, 11;
  3,  4;
  1, 13;
...

Alois P. Heinz, Rows n = 1..5000
Omar E. Pol, Illustration of the divisors of the first 12 positive integers

Row n has length A169695(n).
Row sums give A207376.
Cf. A000005, A000290, A008578, A027750, A033676, A033677, A097448, A161901, A161904.

A207375row[n_] := ArrayPad[#, -Floor[(Length[#] - 1)/2]] & [Divisors[n]];
Array[A207375row, 50] (* Paolo Xausa, Apr 07 2025 *)

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

A210959 Triangle read by rows in which row n lists the divisors of n starting with 1, n, the second smallest divisor of n, the second largest divisor of n, the third smallest divisor of n, the third largest divisor of n, and so on.

Original entry on oeis.org

1, 1, 2, 1, 3, 1, 4, 2, 1, 5, 1, 6, 2, 3, 1, 7, 1, 8, 2, 4, 1, 9, 3, 1, 10, 2, 5, 1, 11, 1, 12, 2, 6, 3, 4, 1, 13, 1, 14, 2, 7, 1, 15, 3, 5, 1, 16, 2, 8, 4, 1, 17, 1, 18, 2, 9, 3, 6, 1, 19, 1, 20, 2, 10, 4, 5, 1, 21, 3, 7, 1, 22, 2, 11, 1, 23, 1, 24

Offset: 1

Omar E. Pol, Jul 29 2012

A two-dimensional arrangement of squares has the property that the number of vertices in row n equals the number of divisors of n. So T(n,k) is represented in the structure as the k-th vertex of row n (see the illustration of initial terms).

			Written as an irregular triangle the sequence begins:
1;
1, 2;
1, 3;
1, 4, 2;
1, 5;
1, 6, 2, 3;
1, 7;
1, 8, 2, 4;
1, 9, 3;
1, 10, 2, 5;
1, 11;
1, 12, 2, 6, 3, 4;

Michel Marcus, Rows 1..1000, flattened
Omar E. Pol, Illustration of initial terms
Index entries for sequences related to sigma(n)

Row n has length A000005(n). Row sums give A000203. Right border gives A033677.

Cf. A027750, A161901, A212119.

row(n) = my(d=divisors(n)); vector(#d, k, if (k % 2, d[(k+1)/2], d[#d-k/2+1])); \\ Michel Marcus, Jun 20 2019

A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.

Original entry on oeis.org

2, 3, 4, 9, 6, 12, 8, 15, 16, 18, 12, 28, 14, 24, 24, 35, 18, 39, 20, 42, 32, 36, 24, 60, 36, 42, 40, 56, 30, 72, 32, 63, 48, 54, 48, 97, 38, 60, 56, 90, 42, 96, 44, 84, 78, 72, 48, 124, 64, 93, 72, 98, 54, 120, 72, 120, 80, 90, 60, 168, 62, 96, 104, 135, 84, 144, 68, 126, 96

Offset: 1

Jason Earls, May 04 2001

Paraphrasing the Jovovic formula: if n is not a square then a(n) = sigma(n), the sum of divisors of n, otherwise a(n) = sigma(n) + sqrt(n). - Omar E. Pol, Jun 23 2009

Row sums of A161901. - Omar E. Pol, Jan 06 2014

			a(4)=9 because pairs of factors are 1*4 and 2*2 and 1+4+2+2=9. a(6)=12 because pairs of factors are 1*6 and 2*3 and 1+6+2+3=12.

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000203, A037213, A060872, A066839, A070038.

A060866 := proc(n)
        numtheory[sigma](n) ;
        if issqr(n) then
                %+sqrt(n) ;
        else
                % ;
        end if;
end proc: # R. J. Mathar, Oct 24 2011

Table[Sum[(i^2 + n) (1 - Ceiling[n/i] + Floor[n/i])/i, {i, Floor[Sqrt[n]]}], {n, 100}] (* Wesley Ivan Hurt, Jul 14 2014 *)
Array[If[IntegerQ@ #2, #3 + #2, #3] & @@ {#, Sqrt@ #, DivisorSigma[1, #]} &, 69] (* Michael De Vlieger, Nov 23 2017 *)

A037213(n) = if(issquare(n,&n),n,0);
A060866(n) = (sigma(n)+A037213(n)); \\ Antti Karttunen, Nov 23 2017, after Jan 25 2003 formula of Vladeta Jovovic

a(n) = A066839(n)+A070038(n) = A000203(n)+A037213(n). G.f.: Sum_{n>0} n*x^n*(x^(n*(n-1))-x^(n^2)+1)/(1-x^n). - Vladeta Jovovic, Jan 25 2003

a(n) = sum_{i=1..floor(sqrt(n))} (n+i^2)*(1-ceiling(n/i)+floor(n/i))/i. - Wesley Ivan Hurt, Jul 14 2014

More terms from Erich Friedman, Jun 03 2001

A161904 Array read by rows in which row n list the divisors of n, but if n is a square then the square root of n appears repeated. Also, the divisors appear as pairs (a,b), where a <= b, such that a*b = n.

Original entry on oeis.org

1, 1, 1, 2, 1, 3, 1, 4, 2, 2, 1, 5, 1, 6, 2, 3, 1, 7, 1, 8, 2, 4, 1, 9, 3, 3, 1, 10, 2, 5, 1, 11, 1, 12, 2, 6, 3, 4, 1, 13, 1, 14, 2, 7, 1, 15, 3, 5, 1, 16, 2, 8, 4, 4, 1, 17, 1, 18, 2, 9, 3, 6, 1, 19, 1, 20, 2, 10, 4, 5, 1, 21, 3, 7

Offset: 1

Omar E. Pol, Jun 23 2009

Row lengths are 2*A038548(n). - R. J. Mathar Jun 28 2009

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

A161839 a(n) = A161835(n)/5.

Original entry on oeis.org

5, 6, 7, 8, 9, 10, 11, 13, 15, 17, 19, 23, 25, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97, 101, 103, 107, 109, 113, 127, 131, 137, 139, 149, 151, 157, 163, 167, 173, 179, 181, 191, 193, 197, 199, 211, 223, 227, 229, 233

Offset: 1

Omar E. Pol, Jun 30 2009

Cf. A018253, A161344, A161345, A161424, A161425, A161835, A161901.

A238288 Triangle read by rows T(n,k), n>=1, k>=1, in which column k lists the positive integers interleaved with k-1 zeros, but starting from 2*k at row k^2.

Original entry on oeis.org

2, 3, 4, 5, 4, 6, 0, 7, 5, 8, 0, 9, 6, 10, 0, 6, 11, 7, 0, 12, 0, 0, 13, 8, 7, 14, 0, 0, 15, 9, 0, 16, 0, 8, 17, 10, 0, 8, 18, 0, 0, 0, 19, 11, 9, 0, 20, 0, 0, 0, 21, 12, 0, 9, 22, 0, 10, 0, 23, 13, 0, 0, 24, 0, 0, 0, 25, 14, 11, 10, 26, 0, 0, 0, 10

Offset: 1

Omar E. Pol, Mar 02 2014

Row sums give A060866.
If n is a square then the row sum gives n^(1/2) + A000203(n) otherwise the row sum gives A000203(n).
Row n has length A000196(n).
Row n has only one positive term iff n is a noncomposite number (A008578).
If the first element of every column is divided by 2 then we have the triangle A237273 whose row sums give A000203.
It appears that there are only eight rows that do not contain zeros. The indices of these rows are in A018253.

			Triangle begins:
2;
3;
4;
5,   4;
6,   0;
7,   5;
8,   0;
9,   6;
10,  0,  6;
11,  7,  0;
12,  0,  0;
13,  8,  7;
14,  0,  0;
15,  9,  0;
16,  0,  8;
17, 10,  0,  8;
18,  0,  0,  0;
19, 11,  9,  0;
20,  0,  0,  0;
21, 12,  0,  9;
22,  0, 10,  0;
23, 13,  0,  0;
24,  0,  0,  0;
25, 14, 11, 10;
26,  0,  0,  0,  10;
27, 15,  0,  0,   0;
28,  0, 12,  0,   0;
29, 16,  0, 11,   0;
30,  0,  0,  0,   0;
31, 17, 13,  0,  11;
...

Index entries for sequences related to sigma(n)

Cf. A000196, A000203, A008578, A018253, A060866, A161901, A196020, A237270, A237273, A238442.

cp's OEIS Frontend

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

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

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A207375 Irregular array read by rows in which row n lists the (one or two) central divisors of n in increasing order.

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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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A210959 Triangle read by rows in which row n lists the divisors of n starting with 1, n, the second smallest divisor of n, the second largest divisor of n, the third smallest divisor of n, the third largest divisor of n, and so on.

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A060866 Sum of (d+d') over all unordered pairs (d,d') with d*d' = n.

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A161904 Array read by rows in which row n list the divisors of n, but if n is a square then the square root of n appears repeated. Also, the divisors appear as pairs (a,b), where a <= b, such that a*b = n.

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