cp's OEIS Frontend

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.

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A269111 a(n) = length of the repeating part of row n of A288097.

Original entry on oeis.org

2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2
Offset: 1

Views

Author

Felix Fröhlich, Feb 19 2016

Keywords

Comments

a(n) + A268479(n) = total number of different terms in the trajectory of p.
a(15) is unknown, since there is no known Wieferich prime in base 47 (cf. Fischer link).
Obviously, a(n) != 1 for all n.
Period length of the repeating part of prime(n)-th row of A281001. - Felix Fröhlich, Jan 14 2017

Examples

			The trajectory of 31 starts 31, 7, 5, 2, 1093, 2, 1093, 2, 1093,  ...., entering a repeating cycle of length 2, so a(11) = 2.

Links

Crossrefs

Cf. A039951, A244550, A252801, A252802, A252812, A268479, A281001, A288097.

Programs

  • Mathematica
    Table[Length@ DeleteCases[Values@ PositionIndex@ NestList[Function[n, Block[{p = 2}, While[! Divisible[n^(p - 1) - 1, p^2], p = NextPrime@ p]; p]], Prime@ n, 12], ?(Length@ # == 1 &)], {n, 12}] (* _Michael De Vlieger, Jun 06 2017, Version 10 *)
  • PARI
    a039951(n) = forprime(p=1, , if(Mod(n, p^2)^(p-1)==1, return(p)))
trajectory(n, terms) = my(v=[n]); while(#v < terms, v=concat(v, a039951(v[#v]))); v
a(n) = my(p=prime(n), i=0, len=2, t=trajectory(p, len), k=#t); while(1, while(k > 1, k--; if(t[k]==t[#t], return(#t-k))); len++; t=trajectory(p, len); k=#t) \\ Felix Fröhlich, Jan 14 2017

Extensions

Definition simplified by Felix Fröhlich, Jun 05 2017

A289899 Primes that are the largest member of a Wieferich cycle.

Original entry on oeis.org

71, 1093, 4871
Offset: 1

Views

Author

Felix Fröhlich, Jul 14 2017

Keywords

Comments

A Wieferich cycle is a repeating cycle in the trajectory of p under successive applications of the map p -> A039951(p), i.e., a part of a row of A288097 repeating indefinitely.
The above cycles could more precisely be called "order-1 Wieferich cycles". A cycle in a row of A281002 could be called an "order-2 Wieferich cycle".
The cycles corresponding to a(1)-a(3) are {3, 11, 71}, {2, 1093} and {83, 4871}, respectively.
The order of the cycle is not to be confused with its length. The order-1 cycle {3, 11, 71} is a cycle of length 3, while the order-1 cycles {2, 1093} and {83, 4871} are cycles of length 2.
Wieferich cycles are special cases of Wieferich tuples (cf. A271100).
a(4) > 20033669 if it exists.

Examples

			71 is a term, since A039951(71) = 3, A039951(3) = 11 and A039951(11) = 71, so {3, 11, 71} is a Wieferich cycle of length 3 and 71 is the largest member of that cycle.

Crossrefs

Cf. A039951, A252801, A252802, A252812, A268479, A269111, A271100, A281002, A288097.

Programs

  • PARI
    leastwieferich(base, bound) = forprime(p=1, bound, if(Mod(base, p^2)^(p-1)==1, return(p))); 0
is(n) = my(v=[leastwieferich(n, n)]); while(1, if(v[#v]==0, return(0), v=concat(v, leastwieferich(v[#v], n))); my(x=#v-1); while(x > 1, if(v[#v]==v[x], if(n==vecmax(v), return(1), return(0))); x--))
forprime(p=1, , if(is(p), print1(p, ", ")))
Showing 1-2 of 2 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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