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

2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 2, 2, 2, 2

Offset: 1

Author

Felix Fröhlich, Feb 19 2016

Keywords

• nonn
• hard
• more

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

• R. Fischer, Thema: Fermatquotient B^(P-1) == 1 (mod P^2)

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