A006556 - OEIS

%I M0175 #29 May 27 2020 20:49:43

%S 2,1,5,2,1,1,1,1,2,12,8,2,1,4,1,1,2,2,9,6,2,2,1,25,3,2,1,1,3,1,17,3,1,

%T 2,2,2,1,4,1,1,2,1,2,2,7,1,2,1,1,34,8,5,1,1,1,54,4,10,2,2,2,2,1,4,3,1,

%U 2,3,11,2,1,2,1,1,1,4,2,2,1,3,2,1,2,2,14,3,1,3,2,2,1,1,1,1,1,10,2,1,6

%N Number of different cycles of digits in the decimal expansions of 1/p, 2/p, ..., (p-1)/p where p = n-th prime different from 2 or 5.

%D J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 162.

%D M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 131.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%H T. D. Noe, <a href="/A006556/b006556.txt">Table of n, a(n) for n=3..1000</a>

%H Victor Meally, <a href="/A006556/a006556.pdf">Letter to N. J. A. Sloane</a>, no date.

%F (p-1)/x, where 10^x = 1 mod p.

%e 1/13=.0769230769..., 2/13=.1538461538..., 3/13= .2307692307..., etc., with 2 different cycles, so a(4) = 2 [13 is the 4th prime different from 2 or 5].

%t Map[(# - 1)/MultiplicativeOrder[10, #] &, {3}~Join~Prime@ Range[4, 101]] (* _Michael De Vlieger_, May 27 2020 *)

%o (PARI) f(p) = (p-1)/znorder(Mod(10, p));

%o lista(nn) = {my(vp=select(x->(10%x), primes(nn))); apply(f, vp);} \\ _Michel Marcus_, May 27 2020

%Y See A048595 and A002371 for the length of the cycles. See also A054471.

%K nonn,easy,base,nice

%O 3,1

%A _N. J. A. Sloane_

%E More terms from _James A. Sellers_, May 24 2000

%E Edited by _Charles R Greathouse IV_, Nov 01 2009