A018266 -id:A018266 - cp's OEIS frontend

A018253 Divisors of 24.

1, 2, 3, 4, 6, 8, 12, 24

Offset: 1

The divisors of 24 greater than 1 are the only positive integers n with the property m^2 == 1 (mod n) for all integer m coprime to n. - Antonio G. Astudillo (afg_astudillo(AT)hotmail.com), Jun 10 2001
Numbers n for which all Dirichlet characters are real. - Benoit Cloitre, Apr 21 2002
These are the numbers n that are divisible by all numbers less than or equal to the square root of n. - Tanya Khovanova, Dec 10 2006 [For a proof, see the Tauvel paper in references. - Bernard Schott, Dec 20 2012]
Also, numbers n such that A160812(n) = 0. - Omar E. Pol, Jun 19 2009
It appears that these are the only positive integers n such that A160812(n) = 0. - Omar E. Pol, Nov 17 2009
24 is a highly composite number: A002182(6)=24. - Reinhard Zumkeller, Jun 21 2010
Chebolu points out that these are exactly the numbers for which the multiplication table of the integers mod n have 1s only on their diagonal, i.e., ab == 1 (mod n) implies a = b (mod n). - Charles R Greathouse IV, Jul 06 2011
It appears that 3, 4, 6, 8, 12, 24 (the divisors >= 3 of 24) are also the only numbers n whose proper non-divisors k are prime numbers if k = d-1 and d divides n. - Omar E. Pol, Sep 23 2011
About the last Pol's comment: I have searched to 10^7 and have found no other terms. - Robert G. Wilson v, Sep 23 2011
Sum_{i=1..8} A000005(a(i))^3 = (Sum_{i=1..8} A000005(a(i)))^2, see Kordemsky in References and Barbeau et al. in Links section. - Bruno Berselli, Dec 29 2014

			Square root of 12 = 3.46... and 1, 2 and 3 divide 12.
From the tenth comment: 1^3 + 2^3 + 2^3 + 3^3 + 4^3 + 4^3 + 6^3 + 8^3 = (1+2+2+3+4+4+6+8)^2 = 900. - _Bruno Berselli_, Dec 28 2014

Harvey Cohn, "Advanced Number Theory", Dover, chap.II, p. 38
Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.
Patrick Tauvel, "Exercices d'algèbre générale et d'arithmétique", Dunod, 2004, exercice 70 page 368.

John Baez, My Favorite Number: 24, The Rankin Lectures (September 19, 2008).
Edward Barbeau and Samer Seraj, Sum of Cubes is Square of Sum, arXiv:1306.5257 [math.NT], 2013.
Paul T. Bateman and Marc E. Low, Prime numbers in arithmetic progressions with difference 24, The American Mathematical Monthly 72:2 (Feb., 1965), pp. 139-143.
Sunil K. Chebolu, What is special about the divisors of 24?, Math. Mag., 85 (2012), 366-372.
M. H. Eggar, A curious property of the integer 24, Math. Gazette 84 (2000), pp. 96-97.
J. C. Lagarias (proposer), Problem 11747, Amer. Math. Monthly, 121 (2014), 83.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group
Index entries for sequences related to divisors of numbers

Cf. A174228, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858-A178864.
Cf. A000005, A158649. - Bruno Berselli, Dec 29 2014
Cf. A303704 (with respect to Astudillo's 2001 comment above).

DivisorsInt(24); # Bruno Berselli, Feb 13 2018

Divisors[24] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(24) \\ Charles R Greathouse IV, Apr 28 2011

divisors(24); # Zerinvary Lajos, Jun 13 2009

a(n) = A161710(n-1). - Reinhard Zumkeller, Jun 21 2009

A018261 Divisors of 48.

Original entry on oeis.org

1, 2, 3, 4, 6, 8, 12, 16, 24, 48

Offset: 1

N. J. A. Sloane

48 is a highly composite number: A002182(8)=48. - Reinhard Zumkeller, Jun 21 2010

These are the orders, without repetition, of the finite subgroups of GL_3(Z); see Conway and Sloane. - Hal M. Switkay, Nov 06 2023

J. H. Conway and N. J. A. Sloane, Low-dimensional lattices. II. Subgroups of GL(n,Z), Proc. R. Soc. Lond. A 419 (1988), 29-68.
Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[48] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(48) \\ Charles R Greathouse IV, Jul 10 2016

A018412 Divisors of 360.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180, 360

Offset: 1

N. J. A. Sloane

Comment from J. Lowell: Regular polygons with n sides in which internal angles have integral number of degrees (n >= 3).
360 is a highly composite number: A002182(13) = 360. - Reinhard Zumkeller, Jun 21 2010
There are 22209 ways to represent 360 as a sum of its distinct divisors (A033630). That's more than any smaller number, hence 360 is in A065218. - Alonso del Arte, Oct 09 2017

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[360] (* Vladimir Joseph Stephan Orlovsky, Feb 21 2012 *)

divisors(360) \\ Charles R Greathouse IV, Jun 21 2017

A018256 Divisors of 36.

Original entry on oeis.org

1, 2, 3, 4, 6, 9, 12, 18, 36

Offset: 1

N. J. A. Sloane

36 is a highly composite number: A002182(7)=36. - Reinhard Zumkeller, Jun 21 2010

Numbers with all prime indices and exponents <= 2. Reversing inequalities gives A062739, strict A353502. - Gus Wiseman, Jun 28 2022

Index entries for sequences related to divisors of numbers

Cf. A018253, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Cf. A003586, A004709, A018338, A018710, A062739, A353502.

Divisors[36] (* Vladimir Joseph Stephan Orlovsky, Feb 16 2012 *)

divisors(36) \\ Charles R Greathouse IV, Jun 21 2017

Intersection of A003586 (3-smooth) and A004709 (cubefree). - Gus Wiseman, Jun 28 2022

A018609 Divisors of 720.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 16, 18, 20, 24, 30, 36, 40, 45, 48, 60, 72, 80, 90, 120, 144, 180, 240, 360, 720

Offset: 1

N. J. A. Sloane

720 is a highly composite number: A002182(14)=720. - Reinhard Zumkeller, Jun 21 2010

Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863, A178864. - Reinhard Zumkeller, Jun 21 2010

Divisors[720] (* Vladimir Joseph Stephan Orlovsky, Nov 19 2010 *)

PARI
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divisors(720)
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A165412 Divisors of 2520.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 12, 14, 15, 18, 20, 21, 24, 28, 30, 35, 36, 40, 42, 45, 56, 60, 63, 70, 72, 84, 90, 105, 120, 126, 140, 168, 180, 210, 252, 280, 315, 360, 420, 504, 630, 840, 1260, 2520

Offset: 1

Reinhard Zumkeller, Sep 17 2009

2520 is the largest and last of most highly composite numbers = A072938(7) = A002182(18) = 2520;
a(A000005(2520)) = a(48) = 2520 is the last term.
A242627(2520*n) = 9. - Reinhard Zumkeller, Jul 16 2014

Eric Weisstein's World of Mathematics, Highly Composite Number
Index entries for sequences related to divisors of numbers

Cf. A018336, A018357, A018388, A018412, A018444, A018490, A018559, and A018676 are subsets.

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018609, A178877, A178878, A178858, A178859, A178860, A178861, A178862, A178863, A178864.

Divisors[2520] (* Wesley Ivan Hurt, Mar 20 2022 *)

divisors(2520) \\ Charles R Greathouse IV, Jun 21 2017

A178864 Divisors of 27720.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 14, 15, 18, 20, 21, 22, 24, 28, 30, 33, 35, 36, 40, 42, 44, 45, 55, 56, 60, 63, 66, 70, 72, 77, 84, 88, 90, 99, 105, 110, 120, 126, 132, 140, 154, 165, 168, 180, 198, 210, 220, 231, 252, 264, 280, 308, 315, 330, 360, 385, 396, 420

Offset: 1

Reinhard Zumkeller, Jun 21 2010

27720 is a highly composite number: A002182(25)=27720;

the sequence is finite with A002183(25)=96 terms: a(96)=27720.

R. Zumkeller, Table of n, a(n) for n = 1..96 (full sequence)
Eric Weisstein's World of Mathematics, Highly Composite Number
Index entries for sequences related to divisors of numbers

Cf. A018253, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858, A178859, A178860, A178861, A178862, A178863.

Divisors[27720] (* Harvey P. Dale, Aug 10 2019 *)

divisors(27720) \\ Charles R Greathouse IV, Jun 21 2017

A018293 Divisors of 120.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

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
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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

cp's OEIS Frontend

A018253 Divisors of 24.

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A018261 Divisors of 48.

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A018412 Divisors of 360.

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A018256 Divisors of 36.

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A018609 Divisors of 720.

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A165412 Divisors of 2520.

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A178864 Divisors of 27720.

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A018293 Divisors of 120.

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