|
Search: id:A007850
|
|
|
| A007850 |
|
Giuga numbers: numbers n such that p divides n/p - 1 for every prime divisor p of n. |
|
+0
5
|
|
| 30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506
(list; graph; listen)
|
|
|
OFFSET
|
1,1
|
|
|
COMMENT
|
There are no other Giuga numbers with <= 8 prime factors. I did an exhaustive search using a PARI script which implemented Borweins and Girgensohn's method for finding n factor solutions given n-2 factors). - Fred Schneider (frederick.william.schneider(AT)gmail.com), 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.)
Giuga numbers are the solution of the differential equation n'=n+1, being n' the arithmetic derivative of n. [From Paolo P. Lava (ppl(AT)spl.at), Nov 16 2009]
|
|
REFERENCES
|
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).
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).
Butske, William; Jaje, Lynda M. and Mayernik, Daniel R., 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.
J.-M. De Koninck, Ces nombres qui nous fascinent, Entry 30, pp 11, Ellipses, Paris 2008.
|
|
LINKS
|
Butske, William; Jaje, Lynda M. and Mayernik, Daniel R., Pdf version
Mersenne Forum, Giuga numbers
Eric Weisstein's World of Mathematics, Link to a section of The World of Mathematics.
Wikipedia, Agoh-Giuga conjecture
|
|
EXAMPLE
|
1910667181420507984555759916338506 = 2 * 3 * 7 * 43 * 1831 * 138683 * 2861051 * 1456230512169437
|
|
CROSSREFS
|
Sequence in context: A049394 A143169 A001201 this_sequence A158580 A097313 A056389
Adjacent sequences: A007847 A007848 A007849 this_sequence A007851 A007852 A007853
|
|
KEYWORD
|
nonn,nice,new
|
|
AUTHOR
|
dborwein(AT)uwo.ca, jborwein(AT)cecm.sfu.ca, pborwein(AT)cecm.sfu.ca and rgirgens(AT)julian.uwo.ca
|
|
EXTENSIONS
|
a(12) from Fred Schneider (frederick.william.schneider(AT)gmail.com), Jul 04 2006
Further references from Fred Schneider (frederick.william.schneider(AT)gmail.com), Aug 19 2006
|
|
|
Search completed in 0.002 seconds
|
