login
This site is supported by donations to The OEIS Foundation.

 

Logo

Please make a donation to keep the OEIS running. We are now in our 55th year. In the past year we added 12000 new sequences and reached 8000 citations (which often say "discovered thanks to the OEIS"). We need to raise money to hire someone to manage submissions, which would reduce the load on our editors and speed up editing.
Other ways to donate

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
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

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. 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

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[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, 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

N. J. A. Sloane

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy. .

Last modified December 6 14:44 EST 2019. Contains 329806 sequences. (Running on oeis4.)