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

 

Logo

Annual Appeal: Today, Nov 11 2014, is the 4th anniversary of the launch of the new OEIS web site. 70,000 sequences have been added in these four years, all edited by volunteers. Please make a donation (tax deductible in the US) to help keep the OEIS running.

Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n. 26
30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

There are no other Giuga numbers with 8 or fewer prime factors. I did an exhaustive search using a PARI script which implemented Borwein and Girgensohn's method for finding n factor solutions given n - 2 factors). - Fred Schneider, Jul 04 2006

One further Giuga number is known with 10 prime factors, namely:

420001794970774706203871150967065663240419575375163060922876441614\

2557211582098432545190323474818 =

2 * 3 * 11 * 23 * 31 * 47059 * 2217342227 * 1729101023519 * 8491659218261819498490029296021 * 58254480569119734123541298976556403 but this may not be the next term. (See the Butske et al. paper.)

Conjecture: Giuga numbers are the solution of the differential equation n' = n + 1, being n' the arithmetic derivative of n. [Paolo P. Lava, Nov 16 2009].

n is a Giuga number if and only if n′ = a*n + 1 for some integer a > 0 (see our preprint in arXiv:1103.2298). - José María Grau Ribas, Mar 19 2011.

A composite number n is a Giuga number if and only if Sum_{i = 1..n} i^phi(n) == -1 (mod n), where phi(n) = A000010(n). - Jonathan Sondow, Jan 03 2014

A composite number n is a Giuga number if and only if Sum_{prime p|n} 1/p = 1/n + an integer. Jonathan Sondow, Jan 08 2014

REFERENCES

J. M. Borwein and E. Wong, A Survey of Results Relating to Giuga's Conjecture on Primality. Vinet, Luc (ed.): Advances in Mathematical Sciences: CRM's 25 Years. Providence, RI: American Mathematical Society. CRM Proc. Lect. Notes. 11, 13-27 (1997).

J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 30, pp 11, Ellipses, Paris 2008.

LINKS

Table of n, a(n) for n=1..12.

D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn, Giuga's Conjecture on Primality, Amer. Math. Monthly 103, No. 1, 40-50 (1996).

William Butske, Lynda M. Jaje, and Daniel R. Mayernik, On the equation Sum_{p | N} 1/p + (1/N)=1, pseudoperfect numbers and perfectly weighted graphs, Math. Comp. 69 (2000), no. 229, 407-420.

Josè Maria Grau and Antonio M. Oller-Marcen, Giuga Numbers and the arithmetic derivative. arXiv:1103.2298

Josè Maria Grau and Antonio M. Oller-Marcen, Generalizing Giuga's conjecture, arXiv:1103.3483

J. M. Grau and A. M. Oller-Marcén, On the congruence sum_{j=1}^{n-1} j^{k(n-1)} == -1 (mod n); k-strong Giuga and k-Carmichael numbers, arXiv preprint arXiv:1311.3522, 2013

Mersenne Forum, Giuga numbers

Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867v1 [math.NT], May 04 2013.

J. Sondow and E. Tsukerman, The p-adic order of power sums, the Erdos-Moser equation, and Bernoulli numbers, arXiv:1401.0322 [math.NT], 2014; see section 4.

Eric Weisstein's World of Mathematics, Giuga Number.

Wikipedia, Agoh-Giuga conjecture

Wikipedia, Giuga number

FORMULA

Sum_{i = 1..a(n)} i^phi(a(n)) == -1 (mod a(n)). - Jonathan Sondow, Jan 03 2014

EXAMPLE

1910667181420507984555759916338506 = 2 * 3 * 7 * 43 * 1831 * 138683 * 2861051 * 1456230512169437

CROSSREFS

Cf. A054377, A216823, A216824, A235137, A235138, A235140, A235363, A236434.

Sequence in context: A049394 A143169 A001201 * A162833 A163208 A163552

Adjacent sequences:  A007847 A007848 A007849 * A007851 A007852 A007853

KEYWORD

nonn,nice,hard,more

AUTHOR

D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn.

EXTENSIONS

a(12) from Fred Schneider, Jul 04 2006

Further references from Fred Schneider, Aug 19 2006

Definition corrected by Jonathan Sondow, Sep 16 2012

STATUS

approved

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

Content is available under The OEIS End-User License Agreement .

Last modified December 22 09:16 EST 2014. Contains 252339 sequences.