A040017 - OEIS

%I #55 Feb 16 2025 08:32:38

%S 3,11,37,101,9091,9901,333667,909091,99990001,999999000001,

%T 9999999900000001,909090909090909091,1111111111111111111,

%U 11111111111111111111111,900900900900990990990991,909090909090909090909090909091

%N Prime 3 followed by unique period primes (the period r of 1/p is not shared with any other prime) of the form A019328(r)/gcd(A019328(r),r) in order (periods r are given in A051627).

%C Prime p=3 is the only known example of a unique period prime such that A019328(r)/gcd(A019328(r),r) = p^k with k > 1 (cf. A323748). It is plausible to assume that no other such prime exists. Under this (unproved) assumption, the current sequence lists all unique period primes in order and represents a sorted version of A007615. - _Max Alekseyev_, Oct 14 2022

%D J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.

%H Robert G. Wilson v, <a href="/A040017/b040017.txt">Table of n, a(n) for n = 1..47</a>

%H Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/UniquePrime.html">Unique prime</a>.

%H C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=62">Unique</a>.

%H Chris K. Caldwell and Harvey Dubner, <a href="https://citeseerx.ist.psu.edu/pdf/b19adc95cabdacbf4bf5b92583fe07eecbd4ea75">Unique-Period Primes</a>, J. Recreational Math., 29:1 (1998) 43-48.

%H Ernest G. Hibbs, <a href="https://www.proquest.com/openview/4012f0286b785cd732c78eb0fc6fce80">Component Interactions of the Prime Numbers</a>, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.

%H Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UniquePrime.html">Unique Prime</a>.

%H Wikipedia, <a href="http://en.wikipedia.org/wiki/Unique_prime">Unique prime</a>.

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

%F For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - _Max Alekseyev_, Oct 14 2022

%e The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.

%t lst = {}; Do[c = Cyclotomic[n, 10]; q = c/GCD[c, n]; If[PrimeQ[q], AppendTo[lst, q]], {n, 62}]; Prepend[Sort[lst], 3] (* _Arkadiusz Wesolowski_, May 13 2012 *)

%Y Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.

%K nonn,base,nice,changed

%O 1,1

%A _Jud McCranie_

%E Missing term a(45) inserted in b-file at the suggestion of _Eric Chen_ by _Max Alekseyev_, Oct 13 2022

%E Edited by _Max Alekseyev_, Oct 14 2022