This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A086004 #20 Jun 08 2025 16:15:42
%S A086004 12917,12919,18911,18913,22907,24907,26903,28901,1088063,1288043,
%T A086004 1408031,1428029,1528019,100083679,100280419,100283849,100483847,
%U A086004 100692793,100880413,101080159,101283839,101683093,101683663,102080149
%N A086004 Primes which remain prime after one and after two and after three applications of the rotate-and-add operation of A086002.
%C A086004 These are the primes of A086003 which in addition remain prime after one additional, third application of the rotate-and-add operation.
%C A086004 Note: Have not yet found any 4-Rotation Cycle Primes.
%C A086004 Conjecture 1: Rotation and addition of primes with even numbers of digits never yields a prime.
%C A086004 Conjecture 2: There are no 5-Rotation Cycle Primes.
%C A086004 [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 A086004 4-Rotation Cycle Primes exist and are listed in A261458. - _Chai Wah Wu_, Aug 20 2015
%H A086004 Chai Wah Wu, <a href="/A086004/b086004.txt">Table of n, a(n) for n = 1..10000</a>
%F A086004 {p in A086003: p+rot(p) in A086003}.
%e A086004 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 A086004 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 A086004 Cf. A086002, A086003.
%K A086004 base,nonn
%O A086004 1,1
%A A086004 _Chuck Seggelin_, Jul 07 2003
%E A086004 Condensed by _R. J. Mathar_, Sep 17 2009