A006556 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.

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, 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, 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

Offset: 3

Author

N. J. A. Sloane

Keywords

• nonn
• easy
• base
• nice

Examples

			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].
		

References

• J. H. Conway and R. K. Guy, The Book of Numbers, Copernicus Press, NY, 1996, p. 162.
• M. Kraitchik, Recherches sur la Théorie des Nombres. Gauthiers-Villars, Paris, Vol. 1, 1924, Vol. 2, 1929, see Vol. 1, p. 131.
• N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

• T. D. Noe, Table of n, a(n) for n=3..1000
• Victor Meally, Letter to N. J. A. Sloane, no date.

Crossrefs

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

Programs

Mathematica

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

PARI

f(p) = (p-1)/znorder(Mod(10, p));
lista(nn) = {my(vp=select(x->(10%x), primes(nn))); apply(f, vp);} \\ Michel Marcus, May 27 2020

Formula

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

Extensions

More terms from James Sellers, May 24 2000

Edited by Charles R Greathouse IV, Nov 01 2009