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A174711
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Composites of the form 2*n^n + 1 = A216147(n).
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0
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9, 55, 513, 6251, 93313, 1647087, 33554433, 774840979, 20000000001, 570623341223, 605750213184507, 22224013651116033, 875787780761718751, 36893488147419103233, 1654480523772673528355, 3956839311320627178247959
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OFFSET
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1,1
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COMMENTS
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If p = n+2 is prime, then p divides 2*n^n + 1. Proof : Let p = n+2 prime. According to the Fermat's theorem, then n^(p-1) ==1 (mod p). Because p-1 = n+1, then n^(n+1) ==1 (mod p), and with n=p-2 ==-2 (mod p), we obtain successively : n*n^n ==1 (mod p), -2*n^n ==1 (mod p), 2*n^n == -1 (mod p) => p divides 2*n^n + 1
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REFERENCES
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T.M. Apostol, Introduction to Analytic Number Theory, Springer-Verlag, 1976
J.M. De Koninck, A. Mercier, 1001 problemes en theorie classique des nombres, Ellipses 2004, p.52
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 2.
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LINKS
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Table of n, a(n) for n=1..16.
C. K. Caldwell, Composite Numbers
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EXAMPLE
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a(2) = 9 = 3^2, a(3) = 55 = 5*11, a(4) = 513 = 3 ^ 3 * 19
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MAPLE
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with(numtheory):for n from 0 to 50 do: x:=2*n^n + 1 : if type(x, prime)=false then print (x):else fi:od:
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CROSSREFS
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Complement of A216148 in A216147. - M. F. Hasler, Sep 02 2012
Sequence in context: A183805 A037578 A096191 * A041148 A114026 A212150
Adjacent sequences: A174708 A174709 A174710 * A174712 A174713 A174714
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KEYWORD
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nonn
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AUTHOR
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Michel Lagneau, Mar 27 2010
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STATUS
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approved
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