%I #102 May 13 2023 10:53:56
%S 1,2,3,4,6,8,12,24
%N Divisors of 24.
%C 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
%C Numbers n for which all Dirichlet characters are real. - _Benoit Cloitre_, Apr 21 2002
%C 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]
%C Also, numbers n such that A160812(n) = 0. - _Omar E. Pol_, Jun 19 2009
%C It appears that these are the only positive integers n such that A160812(n) = 0. - _Omar E. Pol_, Nov 17 2009
%C 24 is a highly composite number: A002182(6)=24. - _Reinhard Zumkeller_, Jun 21 2010
%C 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
%C 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
%C 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
%C 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
%D Harvey Cohn, "Advanced Number Theory", Dover, chap.II, p. 38
%D Boris A. Kordemsky, The Moscow Puzzles: 359 Mathematical Recreations, C. Scribner's Sons (1972), Chapter XIII, Paragraph 349.
%D Patrick Tauvel, "Exercices d'algèbre générale et d'arithmétique", Dunod, 2004, exercice 70 page 368.
%H John Baez, <a href="http://math.ucr.edu/home/baez/numbers/24.pdf">My Favorite Number: 24</a>, The Rankin Lectures (September 19, 2008).
%H Edward Barbeau and Samer Seraj, <a href="http://arxiv.org/abs/1306.5257">Sum of Cubes is Square of Sum</a>, arXiv:1306.5257 [math.NT], 2013.
%H Paul T. Bateman and Marc E. Low, <a href="http://www.jstor.org/stable/2310975">Prime numbers in arithmetic progressions with difference 24</a>, The American Mathematical Monthly 72:2 (Feb., 1965), pp. 139-143.
%H Sunil K. Chebolu, <a href="http://arxiv.org/abs/1104.5052">What is special about the divisors of 24?</a>, Math. Mag., 85 (2012), 366-372.
%H M. H. Eggar, <a href="http://www.jstor.org/stable/3621490">A curious property of the integer 24</a>, Math. Gazette 84 (2000), pp. 96-97.
%H J. C. Lagarias (proposer), <a href="https://www.jstor.org/stable/10.4169/amer.math.monthly.121.01.083">Problem 11747</a>, Amer. Math. Monthly, 121 (2014), 83.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/ModuloMultiplicationGroup.html">Modulo Multiplication Group</a>
%H <a href="/index/Di#divisors">Index entries for sequences related to divisors of numbers</a>
%F a(n) = A161710(n-1). - _Reinhard Zumkeller_, Jun 21 2009
%e Square root of 12 = 3.46... and 1, 2 and 3 divide 12.
%e 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
%t Divisors[24] (* _Vladimir Joseph Stephan Orlovsky_, Feb 16 2012 *)
%o (Sage) divisors(24); # _Zerinvary Lajos_, Jun 13 2009
%o (PARI) divisors(24) \\ _Charles R Greathouse IV_, Apr 28 2011
%o (GAP) DivisorsInt(24); # _Bruno Berselli_, Feb 13 2018
%Y Cf. A174228, A018256, A018261, A018266, A018293, A018321, A018350, A018412, A018609, A018676, A178877, A178878, A165412, A178858-A178864.
%Y Cf. A000005, A158649. - _Bruno Berselli_, Dec 29 2014
%Y Cf. A303704 (with respect to Astudillo's 2001 comment above).
%K nonn,fini,full,easy
%O 1,2
%A _N. J. A. Sloane_