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A007663 Fermat quotients: (2^(p-1)-1)/p, where p=prime(n).
(Formerly M2828)
26
1, 3, 9, 93, 315, 3855, 13797, 182361, 9256395, 34636833, 1857283155, 26817356775, 102280151421, 1497207322929, 84973577874915, 4885260612740877, 18900352534538475, 1101298153654301589, 16628050996019877513 (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 (nickh(AT)qbyte.org), 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, 70.

LINKS

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 [math.NT], 2011-2012.

J. 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.

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

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

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

Cf. A002322, A001917, A096060, A001045.

Sequence in context: A003225 A203104 A237355 * A231212 A185174 A018695

Adjacent sequences:  A007660 A007661 A007662 * A007664 A007665 A007666

KEYWORD

nonn,easy,nice

AUTHOR

N. J. A. Sloane, Sep 19 1994

STATUS

approved

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Last modified December 12 23:01 EST 2018. Contains 318081 sequences. (Running on oeis4.)