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

1, 3, 9, 93, 315, 3855, 13797, 182361, 9256395, 34636833, 1857283155, 26817356775, 102280151421, 1497207322929, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 16628050996019877513, 64689951820132126215, 3825714619033636628817

Offset: 2

Author

N. J. A. Sloane, Sep 19 1994

Keywords

• nonn
• easy
• nice

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.

Crossrefs

Cf. A002322, A001917, A096060, A001045.

Programs

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 *)

PARI

forprime(p=3, 100, print1((2^(p-1)-1)/p ", ")) \\ Satish Bysany, Mar 11 2017

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