

A018253


Divisors of 24.


56




OFFSET

1,2


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 nondivisors k are prime numbers if k = d1 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]


REFERENCES

Harvey Cohn, "Advanced Number Theory", Dover, chap.II, p. 38
J. C. Lagarias (proposer), Problem 11747, Amer. Math. Monthly, 121 (2014), 83.
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

Table of n, a(n) for n=1..8.
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. 139143.
Sunil K. Chebolu, What is special about the divisors of 24?, Math. Mag., 85 (2012), 366372.
M. H. Eggar, A curious property of the integer 24, Math. Gazette 84 (2000), pp. 9697.
Eric Weisstein's World of Mathematics, Modulo Multiplication Group
Index entries for sequences related to divisors of numbers


FORMULA

a(n) = A161710(n1).  Reinhard Zumkeller, Jun 21 2009


EXAMPLE

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]


MATHEMATICA

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


PROG

(Sage) divisors(24); # Zerinvary Lajos, Jun 13 2009
(PARI) divisors(24) \\ Charles R Greathouse IV, Apr 28 2011
(GAP) DivisorsInt(24); # Bruno Berselli, Feb 13 2018


CROSSREFS

Cf. A174228, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858A178864.
Cf. A000005, A158649. [Bruno Berselli, Dec 29 2014]
Sequence in context: A007886 A135108 A018515 * A143417 A018597 A018623
Adjacent sequences: A018250 A018251 A018252 * A018254 A018255 A018256


KEYWORD

nonn,fini,full,easy


AUTHOR

N. J. A. Sloane


STATUS

approved



