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%I #83 Aug 11 2018 15:31:13
%S 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,
%T 6,2,870,2,14322,2,510,2,6,2,1919190,2,6,2,13530,2,1806,2,690,2,282,2,
%U 46410,2,66,2,1590,2,798,2,870,2,354,2,56786730,2,6,2,510,2
%N Denominator of Sum_{p prime, p-1 divides n} 1/p.
%C The GCD of integers x^(n+1)-x, for all integers x. - Roger Cuculiere (cuculier(AT)imaginet.fr), Jan 19 2002
%C 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
%C 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
%C 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
%C 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
%C 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
%C 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
%H T. D. Noe, <a href="/A027760/b027760.txt">Table of n, a(n) for n = 1..10000</a>
%H Thomas Clausen, <a href="http://articles.adsabs.harvard.edu/full/1840AN.....17R.351.">Lehrsatz aus einer Abhandlung Über die Bernoullischen Zahlen</a>, Astr. Nachr. 17 (1840), 351-352. [_Peter Luschny_, Apr 29 2009]
%H S. C. Locke and A. Mandel, <a href="http://www.jstor.org/stable/2321762">Problem E 2901</a>, American Mathematical Monthly 88 (1981), p. 538. <a href="http://www.jstor.org/stable/2975559">Solution</a> in Vol. 90 (1983), pp. 212-213. [Daniel M. Rosenblum (DMRosenblum(AT)world.oberlin.edu), Jul 31 2008]
%H D. M. Rosenblum, <a href="http://www.jstor.org/stable/2689508">Problem 1019</a>, Mathematics Magazine 50 (1977), p. 164. <a href="http://www.jstor.org/stable/2689977">Solution</a> by T. Orloff in Vol. 52 (1979), p. 50.
%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Bernoulli_number">Bernoulli number</a>
%F a(2*k) = A091137(2*k)/A091137(2*k-1). - _Paul Curtz_, Aug 05 2008
%F a(n) = product_{p prime, p-1 divides n}. - _Eric M. Schmidt_, Aug 01 2013
%F a(2n-1) = 2. - _Robert G. Wilson v_, Jul 23 2018
%e 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, ...
%p 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
%t clausen[n_] := Product[i, {i, Select[ Map[ # + 1 &, Divisors[n]], PrimeQ]}]
%t Table[clausen[i], {i, 1, 20}] (* _Peter Luschny_, Apr 29 2009 *)
%t f[n_] := Times @@ Select[Divisors@n + 1, PrimeQ]; Array[f, 56] (* _Robert G. Wilson v_, Apr 25 2012 *)
%o (PARI) a(n)=denominator(sumdiv(n,d,if(isprime(d+1),1/(d+1)))) \\ _Charles R Greathouse IV_, Jul 08 2011
%o (PARI) a(n)=my(pr=1);fordiv(n,d,if(isprime(d+1),pr*=d+1));pr \\ _Charles R Greathouse IV_, Jul 08 2011
%o (Sage)
%o def A027760(n):
%o return mul(filter(lambda s: is_prime(s), map(lambda i: i+1, divisors(n))))
%o [A027760(n) for n in (1..56)] # _Peter Luschny_, May 23 2013
%Y Cf. A002997, A027759, A027642, A141056, A317210.
%K nonn,frac
%O 1,1
%A _N. J. A. Sloane_
%E Formula submitted with A141417 added by _R. J. Mathar_, Nov 17 2010
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