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A055628 Primes p whose period of the reciprocal 1/p is (p-1)/3. 13
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

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.

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

EXAMPLE

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

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 *)

CROSSREFS

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

Sequence in context: A095639 A193143 A098049 * A139643 A139957 A077404

Adjacent sequences:  A055625 A055626 A055627 * A055629 A055630 A055631

KEYWORD

nonn,base

AUTHOR

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

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

STATUS

approved

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Last modified May 16 16:06 EDT 2022. Contains 353706 sequences. (Running on oeis4.)