A308695 a(n) is the minimum positive integer m such that m * 2^(n + 2) + 1 is a prime number which does not divide ((F(n + 2) - 1)^m - 1)/(F(n + 2) - 2), where F(n) is the n-th Fermat number (A000215).

1, 2, 1, 8, 4, 2, 1, 128, 64, 32, 16, 8, 4, 2, 1, 6300, 3150, 26, 13, 579, 1069378, 534689, 10, 5, 387304, 193652, 96826, 48413, 141015, 298082, 149041, 2958, 1479, 51418638746, 25709319373, 20, 10, 5, 6, 3

Offset

0,2

Comments

Note that some terms are obtained by dividing the previous one by 2.

From Jinyuan Wang, Feb 18 2020: (Start)

a(n) is the least m such that q = m*2^(n + 2) + 1 is a prime factor of F(n + 2) - 2. Proof: if r(n + 2)/s(n + 2) = ((F(n + 2) - 1)^m - 1)/(F(n + 2) - 2) is not divisible by q, then q divides s(n + 2) because r(n + 2) is always divisible by q (by Fermat's little theorem). Also note that if F(n + 2) - 1 == 1 (mod q), then r(n + 2)/s(n + 2) = Sum_{i = 0..m-1} A001146(n + 2)^i == m (mod q). In conclusion, prime q = m*2^(n + 2) + 1 does not divide r(n + 2)/s(n + 2) if and only if q divides F(n + 2) - 2 = Product_{i = 0..n + 1} F(i).

a(n) always exists because prime factors of F(n) are of the form k*2^(n + 2) + 1. a(n) is not greater than the smallest such k. (End)

a(40) <= 74327396788321657. - Max Alekseyev, Nov 17 2022

Links

Table of n, a(n) for n=0..39.

Lorenzo Sauras Altuzarra, Some arithmetical problems that are obtained by analyzing proofs and infinite graphs, arXiv:2002.03075 [math.NT], 2020.

Example

2 is the minimum positive integer m such that m * 2^(1 + 2) + 1 is a prime number (note that 2 * 2^(1 + 2) + 1 = 17) which does not divide ((F(1 + 2) - 1)^m - 1)/(F(1 + 2) - 2) (note that ((F(1 + 2) - 1)^2 - 1)/(F(1 + 2) - 2) = 257, which is a prime number).

Maple

A308695:=proc(n)
   local m:
   m:=1:
   while not isprime(m*2^(n+2)+1) or (2^(2^(n+2))-1) mod (m*2^(n+2)+1) != 0 do
      m:=m+1:
   od:
   return m:
end proc:

Mathematica

Array[Block[{m = 1}, While[Nand[PrimeQ[#4], Mod[((#3 - 1)^#1 - 1)/(#3 - 2), #4] != 0] & @@ {m, #, 2^(2^(# + 2)) + 1, m*2^(# + 2) + 1}, m++]; m] &, 14] (* Michael De Vlieger, Feb 14 2020 *)

Prog

(PARI) F(n) = 2^(2^n) + 1;
a(n) = {my(m=1); while (!isprime(p=(m*2^(n+2)+1)) || !((((F(n+2)-1)^m-1)/ (F(n+2)-2)) % p), m++); m; } \\ Michel Marcus, Feb 14 2020
(PARI) a(n) = {my(d=4*2^n, q=1); for(m=1, oo, q+=d; if(ispseudoprime(q) && Mod(2, q)^d==1, return(m))); } \\ Jinyuan Wang, Feb 18 2020

Crossrefs

Cf. A000215 (Fermat numbers), A001146.

Sequence in context: A136225 A341724 A089460 * A278111 A223550 A178102

Adjacent sequences: A308692 A308693 A308694 * A308696 A308697 A308698

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Feb 11 2020

Extensions

a(15)-a(32) from Jinyuan Wang, Feb 18 2020

a(33)-a(39) from Max Alekseyev, Nov 17 2022

Status

approved