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A055628 Primes p for which the period of the reciprocal 1/p is (p-1)/3. 7

%I

%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 for which the 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 Richards, Stephen P., A NUMBER FOR YOUR THOUGHTS, 1982, 1984, Box 501, New Providence, NJ, 07974, ISBN 0-9608224-0-2.

%H T. D. Noe, <a href="/A055628/b055628.txt">Table of n, a(n) for n=1..1000</a>

%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-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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Last modified November 13 10:50 EST 2019. Contains 329093 sequences. (Running on oeis4.)