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a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.
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%I #119 Jan 21 2023 14:28:39

%S 2,1093,5,20771,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,

%T 71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,

%U 11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71,3,11,71

%N a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.

%C a(2) = 1093 since 1093 is the smallest odd Wieferich prime to base 2.

%C a(3) = 5 since 5 is the smallest odd Wieferich prime to base 1093.

%C Subsequence starting at a(5) is periodic with period 3, repeating the terms {3, 11, 71}.

%C Do values for a(1) exist such that the resulting sequence does not eventually become periodic?

%C The following table lists the values for a(1) and the resulting cycles those values produce. An entry of the form x-y in first column means all terms from x up to and including y reach the corresponding cycle. An entry of the form {t_1, t_2, t_3, ..., t_n} in second column means the listed terms form a repeating cycle. Entries in second column without curly braces mean the listed terms are reached in order and the term following the last listed term is unknown. A question mark means no further terms have been found in the resulting trajectory of a(1).

%C a(1) | resulting terms

%C ----------------------------------

%C 2-13, 15-20, | {3, 11, 71}

%C 22-28, 30-40, |

%C 42-46, 48-59, |

%C 62-71, 73-82, |

%C 84-87, 89-118, |

%C 120-132, 134-136,|

%C 138, 140-155, |

%C 157-185, 188, |

%C 190-195, 197-199 |

%C |

%C 14, 41, 60, 137, | 29

%C 196 |

%C |

%C 21, 29, 47, 61, | ?

%C 72, 139, 186-187 |

%C |

%C 83 | {4871, 83}

%C |

%C 88 | 2535619637, 139

%C |

%C 119 | 1741

%C |

%C 133 | 5277179

%C |

%C 156 | 347

%C |

%C 189 | 1847

%C |

%C Notes

%C ------

%C The terms of the cycle reached from 83 correspond to A124121(4) and A124122(4), so those terms form a double Wieferich prime pair.

%H R. Fischer, <a href="http://www.fermatquotient.com/FermatQuotienten/FermQ_Sorg">Thema: Fermatquotient B^(P-1) == 1 (mod P^2)</a>

%H <a href="/index/Rec#order_03">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,1).

%F From _Wesley Ivan Hurt_, Jun 30 2016: (Start)

%F G.f.: x*(2+1093*x+5*x^2+20769*x^3-1090*x^4+6*x^5-20700*x^6) / (1-x^3).

%F a(n) = a(n-3) for n>7.

%F a(n) = (85 - 52*cos(2*n*Pi/3) + 68*sqrt(3)*sin(2*n*Pi/3))/3 for n>4. (End)

%p 2,1093,5,20771,seq(op([3, 11, 71]), n=5..50); # _Wesley Ivan Hurt_, Jun 30 2016

%t Join[{2, 1093, 5, 20771},LinearRecurrence[{0, 0, 1},{3, 11, 71},66]] (* _Ray Chandler_, Aug 25 2015 *)

%o (PARI) i=0; a=2; print1(a, ", "); while(i<100, forprime(p=2, 10^6, if(Mod(a, p^2)^(p-1)==1 && p%2!=0, print1(p, ", "); i++; a=p; break({n=1}))))

%o (Magma) [2, 1093, 5, 20771] cat &cat [[3, 11, 71]^^30]; // _Wesley Ivan Hurt_, Jun 30 2016

%Y Cf. A001220, A124121, A124122, A174422, A244546.

%K nonn,easy

%O 1,1

%A _Felix Fröhlich_, Jun 29 2014