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A007663
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Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).
(Formerly M2828)
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14
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1, 3, 9, 93, 315, 3855, 13797, 182361, 9256395, 34636833, 1857283155, 26817356775, 102280151421, 1497207322929, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 16628050996019877513
(list;
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refs;
listen;
history;
text;
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OFFSET
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2,2
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COMMENTS
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The only terms that are perfect squares are a(2) = 1 and a(4) = 9. - Nick Hobson (nickh(AT)qbyte.org), May 20 2007
Contribution from Jonathan Sondow, Jul 19 2010: (Start)
a(n) == 0 (mod 3) if n > 2, since p = prime(n) > 3
and 0 = (-1)^(p-1)-1 == 2^(p-1)-1 (mod 3). (End)
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REFERENCES
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L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 1, p. 105.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
D. Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, 70.
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LINKS
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T. D. Noe, Table of n, a(n) for n=2..100
Nick Hobson, Fermat squares.
J. Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, in Proceedings of CANT 2011, arXiv:1110.3113
H. S. Vandiver, Fermat's Quotients And Related Arithmetic Functions
H. S. Vandiver, New Types Of Congruences Involving Bernoulli Numbers and Fermat's Quotient
H. S. Vandiver, On Congruences Which Relate The Fermat And Wilson Quotients To The Bernoulli Numbers
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FORMULA
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a(n) = 3*A096060(n) for n>2. a(n) = 3*A001045(prime(n)-1)/prime(n) for n>1. - Alexander Adamchuk, Oct 01 2006
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MAPLE
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A007663:= n-> map (p-> (2^(p-1)-1)/p, ithprime(n)):
seq (A007663(n), n=2..20); # - Jani Melik, Jan 24 2011
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MATHEMATICA
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A007663[n_Integer?Positive]:=(-1+2^(Prime[n]-1))/Prime[n]/; (n>1) [From Enrique Pérez Herrero, Sep 08 2010]
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CROSSREFS
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Cf. A002322, A001917, A096060, A001045.
Sequence in context: A018654 A003225 A203104 * A185174 A018695 A156336
Adjacent sequences: A007660 A007661 A007662 * A007664 A007665 A007666
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KEYWORD
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nonn,easy,nice
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AUTHOR
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N. J. A. Sloane.
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STATUS
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approved
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