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A007850
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Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.
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17
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30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506
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OFFSET
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1,1
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COMMENTS
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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.)
Conjecture: Giuga numbers are the solution of the differential equation n'=n+1, being n' the arithmetic derivative of n. [From 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.
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REFERENCES
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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.
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LINKS
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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
Mersenne Forum, Giuga numbers
Eric Weisstein's World of Mathematics, Giuga Number.
Wikipedia, Agoh-Giuga conjecture
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EXAMPLE
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1910667181420507984555759916338506 = 2 * 3 * 7 * 43 * 1831 * 138683 * 2861051 * 1456230512169437
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CROSSREFS
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Sequence in context: A049394 A143169 A001201 * A162833 A163208 A163552
Adjacent sequences: A007847 A007848 A007849 * A007851 A007852 A007853
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KEYWORD
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nonn,nice,hard,more
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AUTHOR
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D. Borwein, J. M. Borwein, P. B. Borwein and R. Girgensohn.
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EXTENSIONS
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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
Definition corrected by Jonathan Sondow, Sep 16 2012
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STATUS
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approved
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