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Search: id:A055628
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| A055628 |
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Primes p for which the period of the reciprocal 1/p is (p-1)/3. |
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+0
7
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| 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; listen)
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OFFSET
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1,1
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COMMENT
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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.
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REFERENCES
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Richards, Stephen P., A NUMBER FOR YOUR THOUGHTS, 1982, 1984, Box 501, New Providence, NJ, 07974, ISBN 0-9608224-0-2.
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LINKS
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T. D. Noe, Table of n, a(n) for n=1..1000
Makoto Kamada, Factorizations of 11...11 (Repunit).
Index entries for sequences related to decimal expansion of 1/n
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EXAMPLE
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127 has period 42 and (127-1)/3 = 126/3 = 42
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MATHEMATICA
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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 &] (from Robert G. Wilson v Sep 14 2004)
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CROSSREFS
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Cf. A054471, A001914, A001913, A097443, A056157, A056210-A056217, A098680
Sequence in context: A066131 A095639 A098049 this_sequence A139643 A139957 A077404
Adjacent sequences: A055625 A055626 A055627 this_sequence A055629 A055630 A055631
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KEYWORD
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nonn
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AUTHOR
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Don Willard (dwillard(AT)prairie.cc.il.us), Jun 05 2000
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EXTENSIONS
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More terms from Robert G. Wilson v (rgwv(AT)rgwv.com), Aug 02 2000
Edited by njas at the suggestion of Andrew Plewe, May 27 2007
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