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A007117
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a(0) = a(1) = 0; for n >= 2, a(n)*2^(n+2) + 1 is the smallest prime factor of the n-th Fermat number F(n) = 2^(2^n) + 1.
(Formerly M4586)
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9
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0, 0, 1, 8, 1024, 5, 1071, 116503103764643, 1209889024954, 1184, 11131, 39, 7, 82731770, 1784180997819127957596374417642156545110881094717, 9264, 3150, 59251857, 13, 33629
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OFFSET
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0,4
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COMMENTS
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a(14) might need to be corrected if F(14) turns out to have a smaller factor than 116928085873074369829035993834596371340386703423373313. F(20) is composite, but no explicit factor is known. - Jeppe Stig Nielsen, Feb 11 2010
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REFERENCES
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P. Ribenboim, The Book of Prime Number Records. Springer-Verlag, NY, 2nd ed., 1989, p. 71.
H. Riesel, ``Prime numbers and computer methods for factorization,'' Progress in Mathematics, Vol. 57, Birkhauser, Boston, 1985, Chap. 4, see p. 377.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
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LINKS
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FORMULA
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EXAMPLE
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F(2) = 2^(2^2) + 1 = 1*2^4 + 1;
F(3) = 2^(2^3) + 1 = 5*2^5 + 1;
F(4) = 2^(2^4) + 1 = 1024*2^6 + 1;
F(5) = 2^(2^5) + 1 = (5*2^7 + 1) * (52347*2^7 + 1);
F(6) = 2^(2^6) + 1 = (1071*2^8 + 1) * (262814145745*2^8 + 1). (End)
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PROG
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(PARI) a(n) = if(n<2, 0, my(lim=2^(2^n-(n+2))); for(k=1, lim, my(p=k*2^(n+2)+1); if(Mod(2, p)^(2^n)==-1, return(k)))) \\ Jianing Song, Mar 02 2021
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CROSSREFS
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KEYWORD
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hard,nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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