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A007663 Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).
(Formerly M2828)
34
1, 3, 9, 93, 315, 3855, 13797, 182361, 9256395, 34636833, 1857283155, 26817356775, 102280151421, 1497207322929, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 16628050996019877513, 64689951820132126215, 3825714619033636628817 (list; graph; refs; listen; history; text; internal format)
OFFSET
2,2
COMMENTS
The only terms that are squares are a(2) = 1 and a(4) = 9. - Nick Hobson, May 20 2007
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)
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
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
REFERENCES
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, p. 70.
LINKS
Jonathan 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 [math.NT], 2011-2012.
Jonathan Sondow, Lerch Quotients, Lerch Primes, Fermat-Wilson Quotients, and the Wieferich-non-Wilson Primes 2, 3, 14771, Combinatorial and Additive Number Theory, CANT 2011 and 2012, Springer Proc. in Math. & Stat., vol. 101 (2014), pp. 243-255.
Andrei K. Svinin, On some class of sums, arXiv:1610.05387 [math.CO], 2016. See p. 11.
H. S. Vandiver, Fermat's Quotients And Related Arithmetic Functions, PNAS 1945 31 (1) pp. 55-60.
FORMULA
From Alexander Adamchuk, Oct 01 2006: (Start)
a(n) = 3*A096060(n) for n > 2.
a(n) = 3*A001045(prime(n)-1)/prime(n) for n > 1. (End)
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
MAPLE
A007663:= n-> map (p-> (2^(p-1)-1)/p, ithprime(n)):
seq (A007663(n), n=2..20); # Jani Melik, Jan 24 2011
MATHEMATICA
A007663[n_Integer?Positive]:=(-1+2^(Prime[n]-1))/Prime[n]/; (n>1) (* Enrique Pérez Herrero, Sep 08 2010 *)
Table[(2^(n-1)-1)/n, {n, Prime[Range[2, 20]]}] (* Harvey P. Dale, Nov 07 2016 *)
PROG
(PARI)
forprime(p=3, 100, print1((2^(p-1)-1)/p ", ")) \\ Satish Bysany, Mar 11 2017
CROSSREFS
Sequence in context: A003225 A203104 A237355 * A231212 A185174 A018695
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Sep 19 1994
STATUS
approved

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Last modified March 19 07:04 EDT 2024. Contains 370953 sequences. (Running on oeis4.)