A244550 a(n) = first odd Wieferich prime to base a(n-1) for n > 1, with a(1) = 2.

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

Offset: 1

Author

Felix Fröhlich, Jun 29 2014

Keywords

• nonn
• easy

Comments

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

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

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

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

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).

a(1) | resulting terms

----------------------------------

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

22-28, 30-40, |

42-46, 48-59, |

62-71, 73-82, |

84-87, 89-118, |

120-132, 134-136,|

138, 140-155, |

157-185, 188, |

190-195, 197-199 |

|

14, 41, 60, 137, | 29

196 |

|

21, 29, 47, 61, | ?

72, 139, 186-187 |

|

83 | {4871, 83}

|

88 | 2535619637, 139

|

119 | 1741

|

133 | 5277179

|

156 | 347

|

189 | 1847

|

Notes

------

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

Links

• R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)
• Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Crossrefs

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

Programs

Magma

[2, 1093, 5, 20771] cat &cat [[3, 11, 71]^^30]; // Wesley Ivan Hurt, Jun 30 2016

Maple

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

Mathematica

Join[{2, 1093, 5, 20771},LinearRecurrence[{0, 0, 1},{3, 11, 71},66]] (* Ray Chandler, Aug 25 2015 *)

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}))))

Formula

From Wesley Ivan Hurt, Jun 30 2016: (Start)

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

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

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