A191611 Places n where A085392(n+1)-A085392(n) = 1.

3, 7, 8, 31, 127, 8191, 131071, 524287, 2147483647, 2305843009213693951, 618970019642690137449562111, 162259276829213363391578010288127, 170141183460469231731687303715884105727

Offset: 1

Author

Jaroslav Krizek, Jun 09 2011

Keywords

• nonn

Comments

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.

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.

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).]

Examples

			Number 8 is in sequence because the difference between A085392(9)=3 and A085392(8)=2 is 1.
31 is in sequence because the difference between A085392(32)=2 and A085392(31)=1 is 1.
		

Links

• B. M. M. de Weger, Book review, Bull. Am. Math. Soc. 25 (1991) 145-146.

Crossrefs

Cf. A085392.

Programs

PARI

gpd(n) = if (n==1, 1, n/factor(n)[1,1]);
gpf(n) = if (n==1, 1, vecmax(factor(n)[,1]));
f(n) = gpf(gpd(n));
isok(n) = f(n+1) - f(n) == 1; \\ Michel Marcus, Apr 08 2018