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A016114 The smallest representative in a cycle of circular primes, where circular primes are numbers that remain prime under cyclic shifts of digits.

%I A016114 #59 Sep 22 2024 18:20:11
%S A016114 2,3,5,7,11,13,17,37,79,113,197,199,337,1193,3779,11939,19937,193939,
%T A016114 199933,1111111111111111111,11111111111111111111111
%N A016114 The smallest representative in a cycle of circular primes, where circular primes are numbers that remain prime under cyclic shifts of digits.
%C A016114 The next repunit that is prime has 317 digits, all ones.  See A004023. - _Harvey P. Dale_, Mar 22 2012
%C A016114 Only the smallest member of the cyclic shift is listed. See A068652 for all members. - _Chai Wah Wu_, Nov 09 2015
%C A016114 It is highly likely that all circular primes not on the list above are repunits (see Caldwell link). - _Ray Chandler_, May 04 2017
%C A016114 Circular primes are A068652 (numbers that remain prime under cyclic shifts of digits). - _Tanya Khovanova_, Jul 29 2024
%H A016114 K. S. Brown, <a href="http://www.mathpages.com/home/kmath359.htm">On General Palindromic Numbers</a>
%H A016114 Chris K. Caldwell, <a href="https://t5k.org/glossary/page.php?sort=CircularPrime">Circular Primes</a>
%H A016114 Patrick De Geest, <a href="http://www.worldofnumbers.com/circular.htm">Circular Primes</a>
%H A016114 James Grime and Brady Haran, <a href="https://www.youtube.com/watch?v=cmJ18ViCUAI">Absolute Primes</a>, YouTube Numberphile video, 2024.
%H A016114 Harvey Heinz, <a href="http://www.magic-squares.net/primes.htm#Circular Primes">Prime Patterns (Illustration using 19937)</a>
%H A016114 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 A016114 Gianni Sarcone, <a href="http://tangente-mag.com/maths_etonnantes.php?id=3139">Tourbillonnants nombres premiers</a>, Tangente Web Site, No date.
%t A016114 circularPrimeQ[p_] := Module[{d = IntegerDigits[p], ps}, ps = Table[FromDigits[d = RotateLeft[d]], {Length[d]}]; If[p > Min[ps], False, And @@ PrimeQ[ps]]]; Select[Prime[Range[100000]], circularPrimeQ] (* _T. D. Noe_, Mar 22 2012 *)
%t A016114 Union[Select[Union/@((FromDigits/@Table[RotateRight[IntegerDigits[#],n],{n,IntegerLength[ #]}])&/@Prime[Range[20000]]),AllTrue[#,PrimeQ]&]][[All,1]] (* The program generates the first 19 terms of the sequence. *) (* _Harvey P. Dale_, Nov 14 2022 *)
%Y A016114 Sequence includes all prime repunits (A004023). Cf. A003459, A293663.
%Y A016114 For a sequence listing all the prime-yielding cyclic permutations see A068652.
%K A016114 nonn,nice,base
%O A016114 1,1
%A A016114 _Robert G. Wilson v_
%E A016114 One more term from _Lekraj Beedassy_, Nov 07 2002
%E A016114 Name edited by _Tanya Khovanova_, Jul 29 2024