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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). 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.)