A027759 -id:A027759 - cp's OEIS frontend

A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

30, 858, 1722, 66198, 2214408306, 24423128562, 432749205173838, 14737133470010574, 550843391309130318, 244197000982499715087866346, 554079914617070801288578559178, 1910667181420507984555759916338506

Offset: 1

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

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, where n' is 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-1} 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. (In fact, all known Giuga numbers n satisfy Sum_{prime p|n} 1/p = 1/n + 1.) - Jonathan Sondow, Jan 08 2014
The prime factors of a(n) are listed as n-th row of A236434. - M. F. Hasler, Jul 13 2015
Conjecture: let k = a(n) and k be the product of x(n) distinct prime factors where x(n) <= x(n+1). Then, for any even n, n/2 + 2 <= x(n) <= n/2 + 3 and, for any odd n, (n+1)/2 + 2 <= x(n) <= (n+1)/2 + 3. For any n > 1, there are y "old" distinct prime factors o(1)...o(y) such that o(1) = 2, o(2) = 3, and z "new" distinct prime factors n(1)...n(z) such that none of them - unlike the "old" ones - can be a divisor of a(q) while q < n; n(1) > o(y), y = x(n) - z >= 2, 2 <= z <= b where b is either 4, or 1/2*n. - Sergey Pavlov, Feb 24 2017
Conjecture: a composite n is a Giuga number if and only if Sum_{k=1..n-1} k^lambda(n) == -1 (mod n), where lambda(n) = A002322(n). - Thomas Ordowski and Giovanni Resta, Jul 25 2018
A composite number n is a Giuga number if and only if A326690(n) = 1. - Jonathan Sondow, Jul 19 2019
A composite n is a Giuga number if and only if n * A027641(phi(n)) == - A027642(phi(n)) (mod n^2). Note: Euler's phi function A000010 can be replaced by the Carmichael lambda function A002322. - Thomas Ordowski, Jun 07 2020
By von Staudt and Clausen theorem, a composite n is a Giuga number if and only if n * A027759(phi(n)) == A027760(phi(n)) (mod n^2). Note: Euler's phi function can be replaced by the Carmichael lambda function. - Thomas Ordowski, Aug 01 2020

			From _M. F. Hasler_, Jul 13 2015: (Start)
The prime divisors of 30 are {2, 3, 5}, and 2 divides 30/2-1 = 14, 3 divides 30/3-1 = 9, and 5 divides 30/5-1 = 5.
The prime divisors of 858 are {2, 3, 11, 13} and 858/2-1 = 428 is even, 858/3-1 = 285 is divisible by 3, 858/11-1 = 77 is a multiple of 11, and 858/13-1 = 65 = 13*5.
(End)

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

M. A. Alekseyev, J. M. Grau, A. M. Oller-Marcen. Computing solutions to the congruence 1^n + 2^n + ... + n^n == p (mod n). Discrete Applied Mathematics, 2018. doi:10.1016/j.dam.2018.05.022 arXiv:1602.02407 [math.NT], 2016.
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).
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.
José María Grau and Antonio M. Oller-Marcén, Giuga Numbers and the arithmetic derivative., arXiv:1103.2298 [math.NT], 2011; J. Int. Seq. 15 (2012) 12.4.1
José María Grau and Antonio M. Oller-Marcén, Generalizing Giuga's conjecture, arXiv:1103.3483 [math.NT], 2011.
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 [math.NT], 2013.
J. M. Grau, A. M. Oller-Marcén, and D. Sadornil, On µ-Sondow Numbers, arXiv:2111.14211 [math.NT], 2021.
John Machacek, Egyptian Fractions and Prime Power Divisors, arXiv:1706.01008 [math.NT], 2017.
Mersenne Forum, Giuga numbers
Romeo Meštrović, Generalizations of Carmichael numbers I, arXiv:1305.1867 [math.NT], May 04 2013.
R. Mestrovic, On a Congruence Modulo n^3 Involving Two Consecutive Sums of Powers, Journal of Integer Sequences, Vol. 17 (2014), 14.8.4.
J. Sondow and K. MacMillan, Primary pseudoperfect numbers, arithmetic progressions, and the Erdős-Moser equation, Amer. Math. Monthly, 124 (2017) 232-240; arXiv:math/1812.06566 [math.NT], 2018.
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

Cf. A054377, A216823, A216824, A235137, A235138, A235140, A235363, A236434, A326690.

fQ[n_] := AllTrue[First /@ FactorInteger@ n, Divisible[n/# - 1, #] &]; Select[Range@ 100000, CompositeQ@ # && fQ@ # &] (* Michael De Vlieger, Oct 05 2015 *)

is(n)=if(isprime(n), return(0)); my(f=factor(n)[,1]); for(i=1,#f, if((n/f[i])%f[i]!=1, return(0))); n>1 \\ Charles R Greathouse IV, Apr 28 2015

from itertools import count, islice
from sympy import isprime, primefactors
def A007850_gen(startvalue=2): # generator of terms >= startvalue
    return filter(lambda x: not isprime(x) and all((x//p-1) % p == 0 for p in primefactors(x)), count(max(startvalue,2)))
A007850_list = list(islice(A007850_gen(),4)) # Chai Wah Wu, Feb 19 2022

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

a(12) from Fred Schneider, Jul 04 2006
Further references from Fred Schneider, Aug 19 2006
Definition corrected by Jonathan Sondow, Sep 16 2012

A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

Original entry on oeis.org

2, 6, 2, 30, 2, 42, 2, 30, 2, 66, 2, 2730, 2, 6, 2, 510, 2, 798, 2, 330, 2, 138, 2, 2730, 2, 6, 2, 870, 2, 14322, 2, 510, 2, 6, 2, 1919190, 2, 6, 2, 13530, 2, 1806, 2, 690, 2, 282, 2, 46410, 2, 66, 2, 1590, 2, 798, 2, 870, 2, 354, 2, 56786730, 2, 6, 2, 510, 2

Offset: 1

N. J. A. Sloane

The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002
If each x in a ring satisfies x^(n+1)=x, the characteristic of the ring is a divisor of a(n) (Rosenblum 1977). - Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Sep 24 2008
The denominators of the Bernoulli numbers for n>0. B_n sequence begins 1, -1/2, 1/6, 0/2, -1/30, 0/2, 1/42, 0/2, ... This is an alternative version of A027642 suggested by the theorem of Clausen. To add a(0) = 1 has been proposed in A141056. - Peter Luschny, Apr 29 2009
For N > 1, a(n) is the greatest number k such that x*y^n ==y*x^n (mod k) for any integers x and y.  Example: a(19) = 798 because x*y^19 ==y*x^19 (mod 798). - Michel Lagneau, Apr 21 2012
a(n) is the largest k such that b^(n+1) == b (mod k) for every integer b. - Mateusz Szymański, Feb 18 2016, corrected by Thomas Ordowski, Jul 01 2018
When n is even, a(n) is the product of the distinct primes dividing the denominator of zeta(1-n), where zeta(s) is the Riemann zeta function. - Griffin N. Macris, Jun 13 2016
If n+1 is prime, then A002322(a(n)) = n. Composite numbers n+1 such that A002322(a(n)) = n are in A317210. - Max Alekseyev and Thomas Ordowski, Jul 09 2018

			1/2, 5/6, 1/2, 31/30, 1/2, 41/42, 1/2, 31/30, 1/2, 61/66, 1/2, 3421/2730, 1/2, 5/6, 1/2, 557/510, ...

T. D. Noe, Table of n, a(n) for n = 1..10000
Thomas Clausen, Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen, Astr. Nachr. 17 (1840), 351-352. [_Peter Luschny_, Apr 29 2009]
S. C. Locke and A. Mandel, Problem E 2901, American Mathematical Monthly 88 (1981), p. 538. Solution in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]
D. M. Rosenblum, Problem 1019, Mathematics Magazine 50 (1977), p. 164. Solution by T. Orloff in Vol. 52 (1979), p. 50.
Wikipedia, Bernoulli number

Cf. A002997, A027759, A027642, A141056, A317210.

A027760 := proc(n) local s,p; s := 0 ; p := 2; while p <= n+1 do if n mod (p-1) = 0 then s := s+1/p; fi; p := nextprime(p) ; od: denom(s) ; end: # R. J. Mathar, Aug 12 2008

clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
Table[clausen[i], {i, 1, 20}] (* Peter Luschny, Apr 29 2009 *)
f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* Robert G. Wilson v, Apr 25 2012 *)

a(n)=denominator(sumdiv(n,d,if(isprime(d+1),1/(d+1)))) \\ Charles R Greathouse IV, Jul 08 2011

a(n)=my(pr=1);fordiv(n,d,if(isprime(d+1),pr*=d+1));pr \\ Charles R Greathouse IV, Jul 08 2011

def A027760(n):
    return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
[A027760(n) for n in (1..56)]  # Peter Luschny, May 23 2013

a(2*k) = A091137(2*k)/A091137(2*k-1). - Paul Curtz, Aug 05 2008
a(n) = product_{p prime, p-1 divides n}. - Eric M. Schmidt, Aug 01 2013
a(2n-1) = 2. - Robert G. Wilson v, Jul 23 2018

Formula submitted with A141417 added by R. J. Mathar, Nov 17 2010

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A007850 Giuga numbers: composite numbers n such that p divides n/p - 1 for every prime divisor p of n.

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A027760 Denominator of Sum_{p prime, p-1 divides n} 1/p.

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