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A103743
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a(n) = 2^(2^n-1) - 2^n - 1.
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0
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-1, -1, 3, 119, 32751, 2147483615, 9223372036854775743, 170141183460469231731687303715884105599, 57896044618658097711785492504343953926634992332820282019728792003956564819711
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OFFSET
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0,3
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COMMENTS
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It appears that this is not prime for n >= 3? - Posting to Number Theory List by Georges Z., Mar 25 2005
This is indeed true. Let d_n=2^(2^n-1)-2^n-1. Then 2d_n=2^(2^n)-1-(2^(n+1)+1)=Product_{r=0..n-1} F_r-(2^(n+1)+1) where F_r=2^(2^r)+1 is the r-th Fermat number. Write n+1=q*2^r where q is odd, then r<n (since 2^n>n+1 for n>2) and F_r divides 2^(n+1)+1. Therefore F_r divides 2d_n. On the other hand, F_r<=2^(2^(n-1))+1<d_n. So d_n is composite. See my recent survey "Problems and results on covering systems" (Lecture 24) and my paper 43 (joint with M. H. Le). - Zhi-Wei Sun, Mar 25 2005
The next term -- a(9) -- has 154 digits. - Harvey P. Dale, Aug 14 2011
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LINKS
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MATHEMATICA
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PROG
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(PARI) vector(10, n, 2^(2^(n-1)-1)-2^(n-1)-1) \\ Derek Orr, Mar 29 2015
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CROSSREFS
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KEYWORD
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sign,easy
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AUTHOR
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STATUS
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approved
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