A332414 Positive integers r such that A(1,r) = A(2,r - 1) = ... = A(r,1) = 0, where A denotes the function mapping every pair of positive integers (m,n) into 1 if m * 2^(n + 2) + 1 is a prime number dividing F(n + 2) - 2, where F(n) denotes the n-th Fermat number (i.e., F(n) = A000215(n)); and into 0 otherwise.

1, 3, 4, 5, 8, 11, 12, 16, 19, 20, 21, 22, 23, 26, 28, 29, 32, 33, 34, 35, 36, 37, 38, 39, 44, 46, 47, 51, 52, 53, 55, 56, 57, 58, 60, 61, 62, 63, 64, 65, 66, 67, 68, 69, 70, 72, 74, 75, 76, 78, 80, 82, 84, 85, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 99, 100, 101

Offset

1,2

Comments

Note that this sequence is a subsequence of A332416.

Prime q = m*2^(n + 2) + 1 does not divide ((F(n + 2) - 1)^m - 1)/(F(n + 2) - 2) if and only if q divides F(n + 2) - 2 = Product_{i = 0..n + 1} F(i). Direct implication is Theorem 2.26 of my article (see the links) and reciprocal implication is due to Wang (see A308695).

Links

Table of n, a(n) for n=1..67.

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

Example

3 is a term of this sequence, because A(1,3) = A(2,2) = A(3,1) = 0.

Maple

A332414:=proc(n)
   local c, i, k, q, r, v:
   c:=0:
   i:=0:
   r:=1:
   while c < n do
      for k from 0 to r-1 do
         q:=(k+1)*2^(r-k+2)+1:
         if not isprime(q) or (2^(2^(r-k+2)) - 1) mod q != 0 then
            i:=i+1:
         fi:
      od:
      if i = r then
         v:=r:
         c:=c+1:
      fi:
      i:=0:
      r:=r+1:
   od:
   return v:
end proc:

Mathematica

Select[Range@ 29, NoneTrue[Transpose@ {#, Reverse@ #} &@ Range@ #, And[PrimeQ[#4], Mod[((#3 - 1)^#1 - 1)/(#3 - 2), #4] != 0] & @@ {#1, #2, 2^(2^(#2 + 2)) + 1, #1*2^(#2 + 2) + 1} & @@ # &] &] (* Michael De Vlieger, Feb 14 2020 *)

Prog

(PARI) isA(m, t) = ispseudoprime(q=4*m*2^t+1) && Mod(2, q)^(4*2^t)==1;
isok(r) = sum(i=1, r, isA(i, r-i+1)) == 0; \\ Jinyuan Wang, Feb 18 2020

Crossrefs

Cf. A000215 (Fermat numbers), A308695, A332416.

Sequence in context: A101210 A206445 A047599 * A050846 A035538 A039881

Adjacent sequences: A332411 A332412 A332413 * A332415 A332416 A332417

Keyword

nonn

Author

Lorenzo Sauras Altuzarra, Feb 12 2020

Extensions

a(17)-a(67) from Jinyuan Wang, Feb 18 2020

Status

approved