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
Albert H. Beiler, Recreations in the theory of numbers, New York, Dover, (2nd ed.) 1966. See pp. 47, 308.
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
T. D. Noe, Table of n, a(n) for n=2..100
Nick Hobson, Solution to puzzle 158: Fermat squares.
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], 2016-2017. See p. 11.
H. S. Vandiver, Fermat's Quotients And Related Arithmetic Functions, PNAS 1945 31 (1) pp. 55-60.
H. S. Vandiver, New Types Of Congruences Involving Bernoulli Numbers and Fermat's Quotient, PNAS 1948 34 (3) pp. 103-110.
H. S. Vandiver, On Congruences Which Relate The Fermat And Wilson Quotients To The Bernoulli Numbers, PNAS 1949 35 (6) pp. 332-337.
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
KEYWORD
nonn,easy,nice
AUTHOR
N. J. A. Sloane, Sep 19 1994
STATUS
approved