A055628 Primes p whose period of the reciprocal 1/p is (p-1)/3.

103, 127, 139, 331, 349, 421, 457, 463, 607, 661, 673, 691, 739, 829, 967, 1657, 1669, 1699, 1753, 1993, 2011, 2131, 2287, 2647, 2659, 2749, 2953, 3217, 3229, 3583, 3691, 3697, 3739, 3793, 3823, 3931, 4273, 4297, 4513, 4549, 4657, 4903, 4909, 4993, 5011

Offset: 1

Author

Don Willard (dwillard(AT)prairie.cc.il.us), Jun 05 2000

Keywords

• nonn
• base

Comments

Cyclic numbers of the third degree (or third order): the reciprocals of these numbers belong to one of three different cycles. Each cycle has (number-1)/3 digits.

All primes p except 2 or 5 have a reciprocal with period which divides p-1.

Examples

			127 has period 42 and (127-1)/3 = 126/3 = 42.
		

References

• Stephen P. Richards, A Number For Your Thoughts, 1982, 1984, Box 501, New Providence, NJ, 07974, ISBN 0-9608224-0-2.

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Makoto Kamada, Factorizations of 11...11 (Repunit).
• Index entries for sequences related to decimal expansion of 1/n

Crossrefs

Cf. A054471, A001914, A001913, A097443, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.

Programs

Mathematica

LP[ n_Integer ] := (ds = Divisors[ n - 1 ]; Take[ ds, Position[ PowerMod[ 10, ds, n ], 1 ][ [ 1, 1 ] ] ][ [ -1 ] ]); CL[ n_Integer ] := (n - 1)/LP[ n ]; Select[ Range[ 7, 7500 ], PrimeQ[ # ] && CL[ # ] == 3 & ]
f[n_Integer] := Block[{ds = Divisors[n - 1]}, (n - 1)/Take[ ds, Position[ PowerMod[ 10, ds, n], 1] [[1, 1]]] [[ -1]]]; Select[ Prime[ Range[4, 700]], f[ # ] == 3 &] (* Robert G. Wilson v, Sep 14 2004 *)

Extensions

More terms from Robert G. Wilson v, Aug 02 2000

Edited by N. J. A. Sloane at the suggestion of Andrew S. Plewe, May 27 2007