A018253 Divisors of 24.

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

Offset: 1

Author

N. J. A. Sloane

Keywords

• nonn
• fini
• full
• easy

Comments

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

Examples

			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
		

References

• 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.

Links

• 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

Crossrefs

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).

Programs

GAP

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

Mathematica

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

PARI

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

Sage

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

Formula

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