cp's OEIS Frontend

A332416 Positive integers r such that B(1,r) = B(2,r - 1) = ... = B(r,1) = 0, where B 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), where F(n) denotes the n-th Fermat number (i.e., F(n) = A000215(n)); and into 0 otherwise.

%I A332416 #29 May 09 2020 19:13:41
%S A332416 1,3,4,5,6,7,8,11,12,13,14,15,16,17,19,20,21,22,23,24,25,26,28,29,31,
%T A332416 32,33,34,35,36,37,38,39,40,41,42,44,46,47,48,50,51,52,53,54,55,56,57,
%U A332416 58,60,61,62,63,64,65,66,67,68,69,70,71,72,73,74,75,76,78,79
%N A332416 Positive integers r such that B(1,r) = B(2,r - 1) = ... = B(r,1) = 0, where B 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), where F(n) denotes the n-th Fermat number (i.e., F(n) = A000215(n)); and into 0 otherwise.
%C A332416 Note that A332414 is a subsequence of this sequence.
%C A332416 Prime q = m*2^(n + 2) + 1 does not divide ((F(n + 2) - 1)^m - 1)/F(n) if and only if q divides F(n). Direct implication is Theorem 2.24 of my article (see the links). Proof of the reciprocal implication (by Wang): A001146(n) = 2^(2^n) == - 1 (mod q), so ((F(n + 2) - 1)^m - 1)/F(n) = Sum_{i = 0..4*m-1} (-1)^(i+1)*(2^(2^n))^i == -4*m (mod q).
%H A332416 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 A332416 3 is a term of this sequence, because B(1,3) = B(2,2) = B(3,1) = 0.
%p A332416 A332416:=proc(n)
%p A332416    local c, i, k, q, r, v:
%p A332416    c:=0:
%p A332416    i:=0:
%p A332416    r:=1:
%p A332416    while c < n do
%p A332416       for k from 0 to r-1 do
%p A332416          q:=(k+1)*2^(r-k+2)+1:
%p A332416          if not isprime(q) or (2^(2^(r-k)) + 1) mod q != 0 then
%p A332416             i:=i+1:
%p A332416          fi:
%p A332416       od:
%p A332416       if i = r then
%p A332416          v:=r:
%p A332416          c:=c+1:
%p A332416       fi:
%p A332416       i:=0:
%p A332416       r:=r+1:
%p A332416    od:
%p A332416    return v:
%p A332416 end proc:
%t A332416 Select[Range@ 29, NoneTrue[Transpose@ {#, Reverse@ #} &@ Range@ #, And[PrimeQ[#4], Mod[((#3 - 1)^#1 - 1)/(2^(2^#2) + 1), #4] != 0] & @@ {#1, #2, 2^(2^(#2 + 2)) + 1, #1*2^(#2 + 2) + 1} & @@ # &] &] (* _Michael De Vlieger_, Feb 14 2020 *)
%o A332416 (PARI) isB(m, t) = ispseudoprime(q=4*m*2^t+1) && Mod(2, q)^(2^t)==-1;
%o A332416 isok(r) = sum(i=1, r, isB(i, r-i+1)) == 0; \\ _Jinyuan Wang_, Feb 18 2020
%Y A332416 Cf. A000215 (Fermat numbers), A001146, A332414.
%K A332416 nonn
%O A332416 1,2
%A A332416 _Lorenzo Sauras Altuzarra_, Feb 12 2020
%E A332416 a(25)-a(68) from _Jinyuan Wang_, Feb 18 2020