This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A040017 #55 Feb 16 2025 08:32:38
%S A040017 3,11,37,101,9091,9901,333667,909091,99990001,999999000001,
%T A040017 9999999900000001,909090909090909091,1111111111111111111,
%U A040017 11111111111111111111111,900900900900990990990991,909090909090909090909090909091
%N A040017 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 A040017 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 A040017 J.-P. Delahaye, Merveilleux nombres premiers ("Amazing primes"), p. 324, Pour la Science Paris 2000.
%H A040017 Robert G. Wilson v, <a href="/A040017/b040017.txt">Table of n, a(n) for n = 1..47</a>
%H A040017 Chris Caldwell, The Prime Glossary, <a href="https://t5k.org/glossary/xpage/UniquePrime.html">Unique prime</a>.
%H A040017 C. K. Caldwell, "Top Twenty" page, <a href="https://t5k.org/top20/page.php?id=62">Unique</a>.
%H A040017 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 A040017 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 A040017 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/UniquePrime.html">Unique Prime</a>.
%H A040017 Wikipedia, <a href="http://en.wikipedia.org/wiki/Unique_prime">Unique prime</a>.
%H A040017 <a href="/index/1#1overn">Index entries for sequences related to decimal expansion of 1/n</a>
%F A040017 For n >= 2, a(n) = A019328(r) / gcd(A019328(r), r), where r = A051627(n). - _Max Alekseyev_, Oct 14 2022
%e A040017 The decimal expansion of 1/101 is 0.00990099..., having a period of 4 and it is the only prime with that period.
%t A040017 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 A040017 Cf. A007615, A007498, A002371, A048595, A006883, A007732, A051626, A051627, A323748.
%K A040017 nonn,base,nice
%O A040017 1,1
%A A040017 _Jud McCranie_
%E A040017 Missing term a(45) inserted in b-file at the suggestion of _Eric Chen_ by _Max Alekseyev_, Oct 13 2022
%E A040017 Edited by _Max Alekseyev_, Oct 14 2022