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Search: id:A102297
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| A102297 |
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Number of distinct divisors of n+1 where n and n+1 are composite or twin composite numbers. |
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+0
1
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| 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; listen)
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OFFSET
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1,2
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COMMENT
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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.
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EXAMPLE
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For n=8 n+1 = 9 = 3*3 or 1 distinct divisor.
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PROGRAM
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(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) }
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CROSSREFS
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Sequence in context: A014710 A055174 A096369 this_sequence A098398 A131714 A130196
Adjacent sequences: A102294 A102295 A102296 this_sequence A102298 A102299 A102300
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KEYWORD
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easy,nonn
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AUTHOR
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Cino Hilliard (hillcino368(AT)gmail.com), Feb 19 2005
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