A102297 Number of distinct divisors of n+1 where n and n+1 are composite or twin composite numbers.

1, 2, 2, 1, 2, 2, 1, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 2, 2, 2, 2, 2, 2, 2, 1, 2, 3, 2, 3, 2, 2, 2, 3, 1, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 3, 1, 2, 2, 2, 1, 3, 2, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 3, 2, 2, 2, 2, 3, 2, 1, 3, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 2, 2, 2

Offset

1,2

Comments

It is interesting to note that the first such consecutive pair of composite numbers is 8 and 9 which are perfect powers: 2^3 and 3^2. Conjecture: 8 and 9 are the only 2 consecutive composite numbers that are both perfect powers. Or, if x>2, x^m+1 != y^n for all m,n,x,y. Now if we relax the condition that 0 and 1 are not composite, we have 0^m+1 = 1^n for all m,n an infinity of solutions.

Links

Table of n, a(n) for n=1..105.

Example

For n=8 n+1 = 9 = 3*3 or 1 distinct divisor.

Prog

(PARI) f(n) = for(x=1, n, y=composite(x)+1; if(!isprime(y), print1(omega(y)", "))) composite(n) =\The n-th composite number. 1 is def as not prime nor composite. { local(c, x); c=1; x=1; while(c <= n, x++; if(!isprime(x), c++); ); return(x) }

Crossrefs

Sequence in context: A096369 A332289 A365096 * A269570 A243759 A098398

Adjacent sequences: A102294 A102295 A102296 * A102298 A102299 A102300

Keyword

easy,nonn

Author

Cino Hilliard, Feb 19 2005

Status

approved