OFFSET
1,1
COMMENTS
These are the primes of A086003 which in addition remain prime after one additional, third application of the rotate-and-add operation.
Note: Have not yet found any 4-Rotation Cycle Primes.
Conjecture 1: Rotation and addition of primes with even numbers of digits never yields a prime.
Conjecture 2: There are no 5-Rotation Cycle Primes.
[Conjecture 1 is true because rotation for even numbers of the form 10^k*a+b yields 10^k*b+a, so rotation-and-add yields (10^k+1)*(a+b), which obviously contains a divisor A000533. RJM, Sep 17 2009]
4-Rotation Cycle Primes exist and are listed in A261458. - Chai Wah Wu, Aug 20 2015
LINKS
Chai Wah Wu, Table of n, a(n) for n = 1..10000
EXAMPLE
a(1)=12917 is in the sequence because 2-fold rotate-and-add yields the prime 60659 as shown in A086003, and the third application yields 60659+59660 = 120319 which still is prime.
MATHEMATICA
rot[n_]:=Module[{idn=IntegerDigits[n], len}, len=Length[idn]; If[OddQ[ len], FromDigits[ Join[ Take[idn, -Floor[len/2]], {idn[[(len+1)/2]]}, Take[idn, Floor[len/2]]]], FromDigits[ Join[ Take[idn, -len/2], Take[idn, len/2]]]]]; a3rotQ[n_]:=AllTrue[Rest[NestList[ #+rot[ #]&, n, 3]], PrimeQ]; Select[Prime[Range[5880000]], a3rotQ] (* Harvey P. Dale, Apr 26 2022 *)
CROSSREFS
KEYWORD
base,nonn
AUTHOR
Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 07 2003
EXTENSIONS
Condensed by R. J. Mathar, Sep 17 2009
STATUS
approved