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%I #29 Jun 29 2021 05:53:03
%S 103,127,139,331,349,421,457,463,607,661,673,691,739,829,967,1657,
%T 1669,1699,1753,1993,2011,2131,2287,2647,2659,2749,2953,3217,3229,
%U 3583,3691,3697,3739,3793,3823,3931,4273,4297,4513,4549,4657,4903,4909,4993,5011
%N Primes p whose period of the reciprocal 1/p is (p-1)/3.
%C 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.
%C All primes p except 2 or 5 have a reciprocal with period which divides p-1.
%D Stephen P. Richards, A Number For Your Thoughts, 1982, 1984, Box 501, New Providence, NJ, 07974, ISBN 0-9608224-0-2.
%H Amiram Eldar, <a href="/A055628/b055628.txt">Table of n, a(n) for n = 1..10000</a> (terms 1..1000 from T. D. Noe)
%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit/">Factorizations of 11...11 (Repunit)</a>.
%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%e 127 has period 42 and (127-1)/3 = 126/3 = 42.
%t 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 & ]
%t 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 *)
%Y Cf. A054471, A001914, A001913, A097443, A056157, A056210, A056211, A056212, A056213, A056214, A056215, A056216, A056217, A098680.
%K nonn,base
%O 1,1
%A Don Willard (dwillard(AT)prairie.cc.il.us), Jun 05 2000
%E More terms from _Robert G. Wilson v_, Aug 02 2000
%E Edited by _N. J. A. Sloane_ at the suggestion of _Andrew S. Plewe_, May 27 2007
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