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

2, 3, 5, 7, 11, 13, 17, 37, 79, 113, 197, 199, 337, 1193, 3779, 11939, 19937, 193939, 199933, 1111111111111111111, 11111111111111111111111

Offset: 1

Author

Robert G. Wilson v

Keywords

• nonn
• nice
• base

Comments

The next repunit that is prime has 317 digits, all ones. See A004023. - Harvey P. Dale, Mar 22 2012

Only the smallest member of the cyclic shift is listed. See A068652 for all members. - Chai Wah Wu, Nov 09 2015

It is highly likely that all circular primes not on the list above are repunits (see Caldwell link). - Ray Chandler, May 04 2017

Circular primes are A068652 (numbers that remain prime under cyclic shifts of digits). - Tanya Khovanova, Jul 29 2024

Links

• K. S. Brown, On General Palindromic Numbers
• Chris K. Caldwell, Circular Primes
• Patrick De Geest, Circular Primes
• James Grime and Brady Haran, Absolute Primes, YouTube Numberphile video, 2024.
• Harvey Heinz, Prime Patterns (Illustration using 19937)
• Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
• Gianni Sarcone, Tourbillonnants nombres premiers, Tangente Web Site, No date.

Crossrefs

Sequence includes all prime repunits (A004023). Cf. A003459, A293663.

For a sequence listing all the prime-yielding cyclic permutations see A068652.

Programs

Mathematica

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 *)
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 *)

Extensions

One more term from Lekraj Beedassy, Nov 07 2002

Name edited by Tanya Khovanova, Jul 29 2024