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A086004
Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.
5
12917, 12919, 18911, 18913, 22907, 24907, 26903, 28901, 1088063, 1288043, 1408031, 1428029, 1528019, 100083679, 100280419, 100283849, 100483847, 100692793, 100880413, 101080159, 101283839, 101683093, 101683663, 102080149
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
FORMULA
{p in A086003: p+rot(p) in A086003}.
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
Sequence in context: A212079 A162895 A256746 * A090887 A246890 A145333
KEYWORD
base,nonn
AUTHOR
Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 07 2003
EXTENSIONS
Condensed by R. J. Mathar, Sep 17 2009
STATUS
approved