A086004 Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.

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

Author

Chuck Seggelin, Jul 07 2003

Keywords

• base
• nonn

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

Examples

			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.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A086002, A086003.

Programs

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 *)

Formula

{p in A086003: p+rot(p) in A086003}.

Extensions

Condensed by R. J. Mathar, Sep 17 2009