cp's OEIS Frontend

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.

%I A332414 #35 May 09 2020 19:12:42
%S A332414 1,3,4,5,8,11,12,16,19,20,21,22,23,26,28,29,32,33,34,35,36,37,38,39,
%T A332414 44,46,47,51,52,53,55,56,57,58,60,61,62,63,64,65,66,67,68,69,70,72,74,
%U A332414 75,76,78,80,82,84,85,88,89,90,91,92,93,94,95,96,97,99,100,101
%N 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.
%C A332414 Note that this sequence is a subsequence of A332416.
%C A332414 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).
%H A332414 Lorenzo Sauras Altuzarra, <a href="https://arxiv.org/abs/2002.03075">Some arithmetical problems that are obtained by analyzing proofs and infinite graphs</a>, arXiv:2002.03075 [math.NT], 2020.
%e A332414 3 is a term of this sequence, because A(1,3) = A(2,2) = A(3,1) = 0.
%p A332414 A332414:=proc(n)
%p A332414    local c, i, k, q, r, v:
%p A332414    c:=0:
%p A332414    i:=0:
%p A332414    r:=1:
%p A332414    while c < n do
%p A332414       for k from 0 to r-1 do
%p A332414          q:=(k+1)*2^(r-k+2)+1:
%p A332414          if not isprime(q) or (2^(2^(r-k+2)) - 1) mod q != 0 then
%p A332414             i:=i+1:
%p A332414          fi:
%p A332414       od:
%p A332414       if i = r then
%p A332414          v:=r:
%p A332414          c:=c+1:
%p A332414       fi:
%p A332414       i:=0:
%p A332414       r:=r+1:
%p A332414    od:
%p A332414    return v:
%p A332414 end proc:
%t A332414 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 *)
%o A332414 (PARI) isA(m, t) = ispseudoprime(q=4*m*2^t+1) && Mod(2, q)^(4*2^t)==1;
%o A332414 isok(r) = sum(i=1, r, isA(i, r-i+1)) == 0; \\ _Jinyuan Wang_, Feb 18 2020
%Y A332414 Cf. A000215 (Fermat numbers), A308695, A332416.
%K A332414 nonn
%O A332414 1,2
%A A332414 _Lorenzo Sauras Altuzarra_, Feb 12 2020
%E A332414 a(17)-a(67) from _Jinyuan Wang_, Feb 18 2020