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%I #14 Apr 11 2018 23:39:51
%S 3,7,8,31,127,8191,131071,524287,2147483647,2305843009213693951,
%T 618970019642690137449562111,162259276829213363391578010288127,
%U 170141183460469231731687303715884105727
%N Places n where A085392(n+1)-A085392(n) = 1.
%C Likely the union of the Mersenne primes A000668 and the {8}. [Presence of Mersenne primes M is obvious because A085392(M+1)=2 and A085392(M)=1.
%C Absence of other primes p is also clear because A085392(p)=1 and A085392(p+1) >=3 because it contains at least one odd prime factor if not of the Mersenne type.
%C For composite candidates c, we search adjacent c+1 and c with largest noncomposite divisors 2 and 1 or 3 and 2. The first branch enforces c=2 which is immediately discarded. The second branch searches for a power of 3 adjacent to a power of 2, and the solution to this exponential diophantine equation 3^x-2^y=1 is believed to lead only to the 8 (see the Weger review in the link).]
%H B. M. M. de Weger, <a href="http://dx.doi.org/10.1090/S0273-0979-1991-16048-9">Book review</a>, Bull. Am. Math. Soc. 25 (1991) 145-146.
%e Number 8 is in sequence because the difference between A085392(9)=3 and A085392(8)=2 is 1.
%e 31 is in sequence because the difference between A085392(32)=2 and A085392(31)=1 is 1.
%o (PARI) gpd(n) = if (n==1, 1, n/factor(n)[1,1]);
%o gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
%o f(n) = gpf(gpd(n));
%o isok(n) = f(n+1) - f(n) == 1; \\ _Michel Marcus_, Apr 08 2018
%Y Cf. A085392.
%K nonn
%O 1,1
%A _Jaroslav Krizek_, Jun 09 2011
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