A033630 -id:A033630 - cp's OEIS frontend

A018293 Divisors of 120.

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 20, 24, 30, 40, 60, 120

Offset: 1

120 is a highly composite number: A002182(10) = 120. - Reinhard Zumkeller, Jun 21 2010
120 is the first 3-perfect number: A005820(1) = 120. - Michel Marcus, Nov 21 2015
There are 279 ways to partition 120 as a sum of its distinct divisors (see A033630). This is more than any smaller number (hence 120 is listed in A065218). - Alonso del Arte, Oct 12 2017

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864. - Reinhard Zumkeller, Jun 21 2010

Divisors[120] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

PARI
```
divisors(120)
```

A018321 Divisors of 180.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 9, 10, 12, 15, 18, 20, 30, 36, 45, 60, 90, 180

Offset: 1

N. J. A. Sloane

These divisors represent a special case of the "nice angles" discussed at the Geometry Center when bending generating triangles to construct polyhedra (link given below). - Alford Arnold, Apr 16 2000
180 is a highly composite number: A002182(11) = 180. - Reinhard Zumkeller, Jun 21 2010
There are 752 ways to partition 180 as a sum of some of its distinct divisors (see A033630). This is more than any smaller number (hence 180 is listed in A065218). - Alonso del Arte, Sep 20 2017

Geometry Center, Triangle Tilings: Nice Angles
Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018412, A018350, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[180] (* Vladimir Joseph Stephan Orlovsky, Feb 19 2012 *)

divisors(180) \\ Charles R Greathouse IV, Jun 21 2017

A018350 Divisors of 240.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, 240

Offset: 1

N. J. A. Sloane

240 is a highly composite number: A002182(12) = 240. - Reinhard Zumkeller, Jun 21 2010

There are 2158 ways to partition 240 as a sum of some of its distinct divisors (see A033630). This is more than any smaller number (hence 240 is listed in A065218). - Alonso del Arte, Dec 20 2017

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[240] (* Harvey P. Dale, Feb 25 2015 *)

divisors(240) \\ Charles R Greathouse IV, Jun 21 2017

A343382 Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 9, 9, 13, 18, 21, 26, 34, 38, 48, 57, 67, 81, 99, 110, 133, 157, 183, 211, 250, 282, 330, 380, 437, 502, 575, 648, 748, 852, 967, 1095, 1250, 1405, 1597, 1801, 2029, 2287, 2579, 2883, 3245, 3638, 4077, 4557, 5107, 5691, 6356

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either (1) empty, or (2) have smallest part not dividing all the others, or (3) have greatest part not divisible by all the others.

			The a(0) = 1 through a(11) = 9 partitions (empty columns indicated by dots):
  ()  .  .  .  .  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)
                                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)
                                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)
                                                  (5,3,1)  (5,4,1)    (9,2)
                                                           (7,2,1)    (5,4,2)
                                                           (4,3,2,1)  (6,3,2)
                                                                      (6,4,1)
                                                                      (7,3,1)
                                                                      (5,3,2,1)

The first condition alone gives A341450.
The non-strict version is A343346 (Heinz numbers: A343343).
The second condition alone gives A343377.
The strict complement is A343378.
The version for "and" instead of "or" is A343379.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A130689, A200745, A264401, A338470, A339562, A342193, A343337, A343338, A343341, A343342.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A083711 a(n) = A083710(n) - A000041(n-1).

Original entry on oeis.org

1, 1, 1, 2, 1, 4, 1, 5, 3, 7, 1, 14, 1, 13, 8, 20, 1, 33, 1, 40, 14, 44, 1, 85, 6, 79, 25, 117, 1, 181, 1, 196, 45, 233, 17, 389, 1, 387, 80, 545, 1, 750, 1, 839, 165, 1004, 1, 1516, 12, 1612, 234, 2040, 1, 2766, 48, 3142, 388, 3720, 1, 5295, 1, 5606, 663, 7038, 83, 9194, 1, 10379, 1005

Offset: 1

N. J. A. Sloane, Jun 16 2003

Number of integer partitions of n with no 1's with a part dividing all the others. If n > 0, we can assume such a part is the smallest. - Gus Wiseman, Apr 18 2021

			From _Gus Wiseman_, Apr 18 2021: (Start)
The a(6) = 4 through a(12) = 13 partitions:
  (6)      (7)  (8)        (9)      (10)         (11)  (12)
  (3,3)         (4,4)      (6,3)    (5,5)              (6,6)
  (4,2)         (6,2)      (3,3,3)  (8,2)              (8,4)
  (2,2,2)       (4,2,2)             (4,4,2)            (9,3)
                (2,2,2,2)           (6,2,2)            (10,2)
                                    (4,2,2,2)          (4,4,4)
                                    (2,2,2,2,2)        (6,3,3)
                                                       (6,4,2)
                                                       (8,2,2)
                                                       (3,3,3,3)
                                                       (4,4,2,2)
                                                       (6,2,2,2)
                                                       (4,2,2,2,2)
                                                       (2,2,2,2,2,2)
(End)

L. M. Chawla, M. O. Levan and J. E. Maxfield, On a restricted partition function and its tables, J. Natur. Sci. and Math., 12 (1972), 95-101.

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

Allowing 1's gives A083710.
The strict case is A098965.
The complement (except also without 1's) is counted by A338470.
The dual version is A339619.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A001787, A001792, A015723, A097986, A098743, A130689, A130714, A264401, A339563, A342193.

with(combinat): with(numtheory): a := proc(n) c := 0: l := sort(convert(divisors(n), list)): for i from 1 to nops(l)-1 do c := c+numbpart(l[i]-1) od: RETURN(c): end: for j from 2 to 100 do printf(`%d,`,a(j)) od: # James Sellers, Jun 21 2003
# second Maple program:
a:= n-> max(1, add(combinat[numbpart](d-1), d=numtheory[divisors](n) minus {n})):
seq(a(n), n=1..69);  # Alois P. Heinz, Feb 15 2023

a[n_] := If[n==1, 1, Sum[PartitionsP[d-1], {d, Most@Divisors[n]}]];
Table[a[n], {n, 1, 100}] (* Jean-François Alcover, Feb 15 2023 *)

a(n) = Sum_{ d|n, dA000041(d-1).

More terms from James Sellers, Jun 21 2003

A343377 Number of strict integer partitions of n with no part divisible by all the others.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 1, 2, 3, 4, 6, 8, 9, 13, 18, 21, 26, 32, 38, 47, 57, 66, 80, 95, 110, 132, 157, 181, 211, 246, 282, 327, 379, 435, 500, 570, 648, 743, 849, 963, 1094, 1241, 1404, 1592, 1799, 2025, 2282, 2568, 2882, 3239, 3634, 4066, 4554, 5094, 5686, 6346

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part not divisible by all the others.

			The a(5) = 1 through a(12) = 9 partitions:
  (3,2)  (3,2,1)  (4,3)  (5,3)    (5,4)    (6,4)      (6,5)      (7,5)
                  (5,2)  (4,3,1)  (7,2)    (7,3)      (7,4)      (5,4,3)
                         (5,2,1)  (4,3,2)  (5,3,2)    (8,3)      (6,4,2)
                                  (5,3,1)  (5,4,1)    (9,2)      (6,5,1)
                                           (7,2,1)    (5,4,2)    (7,3,2)
                                           (4,3,2,1)  (6,4,1)    (7,4,1)
                                                      (7,3,1)    (8,3,1)
                                                      (5,3,2,1)  (9,2,1)
                                                                 (5,4,2,1)

The dual strict complement is A097986.
The dual version is A341450.
The non-strict version is A343341 (Heinz numbers: A343337).
The strict complement is counted by A343347.
The case with smallest part not divisible by all the others is A343379.
The case with smallest part divisible by all the others is A343381.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A130689, A200745, A264401, A338470, A339562, A343338, A343342, A343345, A343346, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

Original entry on oeis.org

1, 0, 0, 0, 0, 1, 0, 2, 1, 3, 3, 5, 3, 9, 9, 12, 12, 18, 18, 27, 27, 36, 41, 51, 51, 73, 80, 96, 105, 132, 137, 177, 188, 230, 253, 303, 320, 398, 431, 508, 550, 659, 705, 847, 913, 1063, 1165, 1359, 1452, 1716, 1856, 2134, 2329, 2688, 2894, 3345, 3622, 4133

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are either empty, or (1) have smallest part not dividing all the others and (2) have greatest part not divisible by all the others.

			The a(5) = 1 through a(13) = 9 partitions (empty column indicated by dot):
  (3,2)  .  (4,3)  (5,3)  (5,4)    (6,4)    (6,5)    (7,5)    (7,6)
            (5,2)         (7,2)    (7,3)    (7,4)    (5,4,3)  (8,5)
                          (4,3,2)  (5,3,2)  (8,3)    (7,3,2)  (9,4)
                                            (9,2)             (10,3)
                                            (5,4,2)           (11,2)
                                                              (6,4,3)
                                                              (6,5,2)
                                                              (7,4,2)
                                                              (8,3,2)

The first condition alone gives A341450.
The non-strict version is A343342  (Heinz numbers: A343338).
The second condition alone gives A343377.
The opposite version is A343378.
The half-opposite versions are A343380 and A343381.
The version for "or" instead of "and" is A343382.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A083710, A097986, A200745, A264401, A338470, A339562, A342193, A343337, A343341, A343343, A343346, A343347.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&!And@@IntegerQ/@(#/Min@@#)&&!And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

The Heinz numbers for the non-strict version are A343338 = A342193 /\ A343337.

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

Original entry on oeis.org

0, 1, 1, 1, 1, 2, 1, 2, 2, 2, 1, 5, 1, 3, 3, 5, 1, 7, 1, 8, 4, 6, 1, 15, 2, 9, 5, 14, 1, 22, 1, 20, 7, 18, 4, 36, 1, 26, 10, 40, 1, 51, 1, 48, 18, 49, 1, 86, 3, 73, 19, 86, 1, 117, 7, 120, 27, 120, 1, 196, 1, 160, 42, 201, 10, 259, 1, 258, 50, 292, 1, 407, 1, 357, 81, 431, 8, 548, 1, 577

Offset: 1

Vladeta Jovovic, Oct 23 2004

If n > 0, we can assume this part is the smallest. - Gus Wiseman, Apr 18 2021

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

The non-strict version with 1's allowed is A083710.
The non-strict version is A083711.
The version with 1's allowed is A097986.
The Heinz numbers of these partitions are the odd terms of A339563.
The non-strict dual is A339619.
The strict complement is counted by A341450.
A000005 counts divisors.
A000041 counts partitions.
A000070 counts partitions with a selected part.
A006128 counts partitions with a selected position.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
Cf. A130689, A130714, A200745, A264401, A343345, A343347, A343378, A343381.

Take[ CoefficientList[ Expand[ Sum[x^k*Product[1 + x^(k*i), {i, 2, 92}], {k, 2, 92}]], x], {2, 81}] (* Robert G. Wilson v, Nov 01 2004 *)
Table[If[n==0,0,Length[Select[IntegerPartitions[n],!MemberQ[#,1]&&UnsameQ@@#&&And@@IntegerQ/@(#/Min@@#)&]]],{n,0,30}] (* Gus Wiseman, Apr 18 2021 *)

a(n) = Sum_{d|n, dA025147(d-1).

G.f.: Sum_{k>=2} (x^k*Product_{i>=2}(1 + x^(k*i))).

More terms from Robert G. Wilson v, Nov 01 2004

Name shortened by Gus Wiseman, Apr 23 2021

A343347 Number of strict integer partitions of n with a part divisible by all the others.

Original entry on oeis.org

1, 1, 1, 2, 2, 2, 3, 3, 3, 4, 4, 4, 6, 5, 4, 6, 6, 6, 8, 7, 7, 10, 9, 9, 12, 10, 8, 11, 11, 10, 14, 13, 11, 13, 12, 15, 20, 17, 15, 19, 19, 19, 22, 18, 17, 23, 22, 22, 28, 25, 24, 31, 28, 26, 32, 32, 30, 34, 32, 29, 37, 33, 27, 36, 33, 34, 44, 38, 36, 45, 45

Offset: 0

Gus Wiseman, Apr 16 2021

nonn

Alternative name: Number of strict integer partitions of n that are empty or have greatest part divisible by all the others.

			The a(1) = 1 through a(15) = 6 partitions (A..F = 10..15):
  1  2  3   4   5   6   7    8   9    A    B    C     D    E    F
        21  31  41  42  61   62  63   82   A1   84    C1   C2   A5
                    51  421  71  81   91   632  93    841  D1   C3
                                 621  631  821  A2    931  842  E1
                                                B1    A21       C21
                                                6321            8421

Andrew Howroyd, Table of n, a(n) for n = 0..1000

The dual version is A097986 (non-strict: A083710).
The non-strict version is A130689 (Heinz numbers: complement of A343337).
The strict complement is counted by A343377.
The case with smallest part divisible by all the others is A343378.
The case with smallest part not divisible by all the others is A343380.
A000005 counts divisors.
A000009 counts strict partitions.
A000070 counts partitions with a selected part.
A015723 counts strict partitions with a selected part.
A018818 counts partitions into divisors (strict: A033630).
A167865 counts strict chains of divisors > 1 summing to n.
A339564 counts factorizations with a selected factor.
Cf. A064410, A098743, A200745, A264401, A339563, A341450, A343341, A343344, A343379, A343382.

Table[Length[Select[IntegerPartitions[n],#=={}||UnsameQ@@#&&And@@IntegerQ/@(Max@@#/#)&]],{n,0,30}]

seq(n)={Vec(1 + sum(m=1, n, my(u=divisors(m)); x^m*prod(i=1, #u-1, 1 + x^u[i] + O(x^(n-m+1)))))} \\ Andrew Howroyd, Apr 17 2021

G.f.: 1 + Sum_{k>0} (x^k/(1 + x^k))*Product_{d|k} (1 + x^d). - Andrew Howroyd, Apr 17 2021

A065218 Consider the subsets of proper divisors of a number that sum to the number. These are numbers that set a record number of such subsets.

Original entry on oeis.org

1, 6, 12, 24, 36, 48, 60, 120, 180, 240, 360, 720, 840, 1260, 1680, 2520, 5040, 7560, 10080, 15120, 20160, 25200, 27720, 45360, 50400, 55440, 83160, 110880, 166320, 221760, 277200, 332640, 498960, 554400, 665280, 720720, 831600, 1081080, 1441440

Offset: 1

Jud McCranie, Oct 21 2001

nonn

Indices of records in A065205 and A033630. The corresponding records (number of subsets) are in A065219.

This sequence is not a subset of A002182: 831600 belongs to this sequence but not A002182.

			Proper divisors of 12 are {1, 2, 3, 4, 6}. Two subsets of this sum to 12: {2, 4, 6} and {1, 2, 3, 6} - more than any smaller number, so 12 is in the sequence.

Michael De Vlieger, Correlation of A065218 and A065219.

Cf. A065219, A064771, A063205, A005835, A002182, A004394.

With[{s = Table[-1 + SeriesCoefficient[Series[Times @@ ((1 + z^#) & /@ Divisors[n]), {z, 0, n}], n], {n, 2520}]}, FirstPosition[s, #][[1]] & /@ Union@ FoldList[Max, s]] (* Michael De Vlieger, Oct 10 2017 *)

More terms from Franklin T. Adams-Watters, Nov 27 2006
Edited and extended by Max Alekseyev, May 29 2009
Offset changed by Andrey Zabolotskiy, Oct 10 2017

cp's OEIS Frontend

A018293 Divisors of 120.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A018321 Divisors of 180.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A018350 Divisors of 240.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Mathematica

PARI

A343382 Number of strict integer partitions of n with either (1) no part dividing all the others or (2) no part divisible by all the others.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A083711 a(n) = A083710(n) - A000041(n-1).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Maple

Mathematica

Formula

Extensions

A343377 Number of strict integer partitions of n with no part divisible by all the others.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A343379 Number of strict integer partitions of n with no part dividing or divisible by all the other parts.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

Formula

A098965 Number of integer partitions of n into distinct parts > 1 with a part dividing all the other parts.

Views

Author

Keywords

Comments

Links

Crossrefs