A086004 - OEIS

%I #19 Apr 26 2022 14:02:09

%S 12917,12919,18911,18913,22907,24907,26903,28901,1088063,1288043,

%T 1408031,1428029,1528019,100083679,100280419,100283849,100483847,

%U 100692793,100880413,101080159,101283839,101683093,101683663,102080149

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

%C These are the primes of A086003 which in addition remain prime after one additional, third application of the rotate-and-add operation.

%C Note: Have not yet found any 4-Rotation Cycle Primes.

%C Conjecture 1: Rotation and addition of primes with even numbers of digits never yields a prime.

%C Conjecture 2: There are no 5-Rotation Cycle Primes.

%C [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]

%C 4-Rotation Cycle Primes exist and are listed in A261458. - _Chai Wah Wu_, Aug 20 2015

%H Chai Wah Wu, <a href="/A086004/b086004.txt">Table of n, a(n) for n = 1..10000</a>

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

%e 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.

%t 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 *)

%Y Cf. A086002, A086003.

%K base,nonn

%O 1,1

%A Chuck Seggelin (barkeep(AT)plastereddragon.com), Jul 07 2003

%E Condensed by _R. J. Mathar_, Sep 17 2009