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 A018253 Divisors of 24. 58
 1, 2, 3, 4, 6, 8, 12, 24 (list; graph; refs; listen; history; text; internal format)
 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 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] 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 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 FORMULA a(n) = A161710(n-1). - 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 (* 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, A178858-A178864. 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 STATUS approved

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Last modified October 18 14:52 EDT 2019. Contains 328161 sequences. (Running on oeis4.)