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A007615 Primes with unique period length (the periods are given in A007498).
(Formerly M2890)
8

%I M2890

%S 3,11,37,101,333667,9091,9901,909091,1111111111111111111,

%T 11111111111111111111111,99990001,999999000001,909090909090909091,

%U 900900900900990990990991,9999999900000001,909090909090909090909090909091,900900900900900900900900900900990990990990990990990990990991

%N Primes with unique period length (the periods are given in A007498).

%C Additional terms are Phi(n,10)/gcd(n,Phi(n,10)) for the n in A007498, where Phi(n,10) is the n-th cyclotomic polynomial evaluated at 10.

%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

%D Samuel Yates, Period Lengths of Exactly One or Two Prime Numbers, J. Rec. Math., 18 (1985), 22-24.

%H Ray Chandler, <a href="/A007615/b007615.txt">Table of n, a(n) for n = 1..31</a> (first 25 terms from T. D. Noe)

%H C. K. Caldwell, The Prime Glossary, <a href="http://primes.utm.edu/glossary/page.php?sort=UniquePrime">unique prime</a>

%H Makoto Kamada, <a href="https://stdkmd.net/nrr/repunit/phin10.htm">Factorizations of Phi_n(10)</a>

%H <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>

%F a(n) = A061075(A007498(n)). [_Max Alekseyev_, Oct 16 2010]

%F a(n) = A006530(A019328(A007498(n))). - _Ray Chandler_, May 10 2017

%e 3 is the only prime p such that decimal expansion of 1/p has (nontrivial) period exactly 1.

%t nmax = 50; periods = Reap[ Do[ p = Cyclotomic[n, 10] / GCD[n, Cyclotomic[n, 10]]; If[ PrimeQ[p], Sow[n]], {n, 1, nmax}]][[2, 1]]; Cyclotomic[#, 10] / GCD[#, Cyclotomic[#, 10]]& /@ periods // Prepend[#, 3]& (* _Jean-Fran├žois Alcover_, Mar 28 2013 *)

%Y Cf. A007498, A040017, A002371, A048595, A006883, A007732, A051626, A061075, A006530, A019328.

%K nonn,nice,easy,base

%O 1,1

%A _N. J. A. Sloane_, _Robert G. Wilson v_, _Mira Bernstein_

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Last modified May 31 22:45 EDT 2020. Contains 334756 sequences. (Running on oeis4.)