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A007663 Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).

%I A007663 M2828 #104 Dec 29 2024 16:10:46
%S A007663 1,3,9,93,315,3855,13797,182361,9256395,34636833,1857283155,
%T A007663 26817356775,102280151421,1497207322929,84973577874915,
%U A007663 4885260612740877,18900352534538475,1101298153654301589,16628050996019877513,64689951820132126215,3825714619033636628817
%N A007663 Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).
%C A007663 The only terms that are squares are a(2) = 1 and a(4) = 9. - _Nick Hobson_, May 20 2007
%C A007663 From _Jonathan Sondow_, Jul 19 2010: (Start)
%C A007663 a(n) == 0 (mod 3) if n > 2, since p = prime(n) > 3
%C A007663 and 0 = (-1)^(p-1)-1 == 2^(p-1)-1 (mod 3). (End)
%C A007663 p is in A001220 if and only if p | (2^(p-1)-1)/p, i.e., a(n) is divisible by prime(n). - _Felix Fröhlich_, Jun 20 2014
%C A007663 In general, every prime p that is 1 mod q-1 will create a numerator that is 0 mod q via Fermat's Little Theorem, meaning every p with this property (except q) will have a Fermat quotient divisible by q. - _Roderick MacPhee_, May 12 2017
%D A007663 Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See pp. 47, 308.
%D A007663 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.
%D A007663 N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%D A007663 D. Wells, The Penguin Dictionary of Curious and Interesting Numbers, Penguin Books, NY, 1986, p. 70.
%H A007663 T. D. Noe, <a href="/A007663/b007663.txt">Table of n, a(n) for n=2..100</a>
%H A007663 Nick Hobson, <a href="https://web.archive.org/web/20160328125436/http://qbyte.org/puzzles/p158s.html">Solution to puzzle 158: Fermat squares</a>.
%H A007663 Jonathan Sondow, <a href="http://arxiv.org/abs/1110.3113">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, in Proceedings of CANT 2011, arXiv:1110.3113 [math.NT], 2011-2012.
%H A007663 Jonathan Sondow, <a href="https://doi.org/10.1007/978-1-4939-1601-6_17">Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771</a>, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
%H A007663 Andrei K. Svinin, <a href="https://arxiv.org/abs/1610.05387">On some class of sums</a>, arXiv:1610.05387 [math.CO], 2016-2017. See p. 11.
%H A007663 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/31/1/55.pdf">Fermat's Quotients And Related Arithmetic Functions</a>, PNAS 1945 31 (1) pp. 55-60.
%H A007663 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/34/3/103.pdf">New Types Of Congruences Involving Bernoulli Numbers and Fermat's Quotient</a>, PNAS 1948 34 (3) pp. 103-110.
%H A007663 H. S. Vandiver, <a href="http://www.pnas.org/cgi/reprint/35/6/332.pdf">On Congruences Which Relate The Fermat And Wilson Quotients To The Bernoulli Numbers</a>, PNAS 1949 35 (6) pp. 332-337.
%F A007663 From _Alexander Adamchuk_, Oct 01 2006: (Start)
%F A007663 a(n) = 3*A096060(n) for n > 2.
%F A007663 a(n) = 3*A001045(prime(n)-1)/prime(n) for n > 1. (End)
%F A007663 a(n) = Sum_{i=0..(p-3)/2} 2^i*(p-i-2)!/((i+1)!*(p-2*(i+1))!) where p = prime(n), for n >= 2. - _Vladimir Pletser_, Jan 26 2023
%p A007663 A007663:= n-> map (p-> (2^(p-1)-1)/p, ithprime(n)):
%p A007663 seq (A007663(n), n=2..20); # _Jani Melik_, Jan 24 2011
%t A007663 A007663[n_Integer?Positive]:=(-1+2^(Prime[n]-1))/Prime[n]/;(n>1) (* _Enrique Pérez Herrero_, Sep 08 2010 *)
%t A007663 Table[(2^(n-1)-1)/n,{n,Prime[Range[2,20]]}] (* _Harvey P. Dale_, Nov 07 2016 *)
%o A007663 (PARI)
%o A007663 forprime(p=3, 100, print1((2^(p-1)-1)/p ", ")) \\ _Satish Bysany_, Mar 11 2017
%Y A007663 Cf. A002322, A001917, A096060, A001045.
%K A007663 nonn,easy,nice
%O A007663 2,2
%A A007663 _N. J. A. Sloane_, Sep 19 1994