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. (Formerly M0175)

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

3,1

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.

Formula

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

Example

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].

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A363739 A257459 A140879 * A108790 A355914 A117941

Adjacent sequences: A006553 A006554 A006555 * A006557 A006558 A006559

Keyword

nonn,easy,base,nice

Author

N. J. A. Sloane

Extensions

More terms from James A. Sellers, May 24 2000

Edited by Charles R Greathouse IV, Nov 01 2009

Status

approved