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 A102297 Number of distinct divisors of n+1 where n and n+1 are composite or twin composite numbers. 0
 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 (list; graph; refs; listen; history; text; internal format)
 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 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: A055174 A096369 A332289 * A269570 A243759 A098398 Adjacent sequences:  A102294 A102295 A102296 * A102298 A102299 A102300 KEYWORD easy,nonn AUTHOR Cino Hilliard, Feb 19 2005 STATUS approved

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Last modified January 23 21:06 EST 2022. Contains 350515 sequences. (Running on oeis4.)