A284170 Array read by antidiagonals: T(i,j) is the largest prime in the sequence defined by a(1) = prime(i), a(2) = prime(j), a(n) = A006530(a(n-1)+a(n-2)+1) for n>=3, or 0 if that sequence contains arbitrarily large primes.

5, 43, 43, 5, 43, 5, 7, 43, 43, 43, 43, 41, 131, 43, 43, 13, 43, 43, 43, 41, 13, 17, 43, 41, 43, 131, 43, 137, 43, 43, 131, 43, 43, 43, 43, 43, 23, 43, 137, 43, 131, 43, 41, 67, 151, 29, 43, 131, 43, 41, 131, 137, 131, 43, 29, 137, 41, 137, 41, 151, 43, 131, 43, 137, 73, 43, 37, 43, 43, 131, 43, 47

Offset

1,1

Comments

Conjecture: the sequence always eventually repeats, so T(i,j) > 0.

Links

Robert Israel, Table of n, a(n) for n = 1..14196 (first 168 antidiagonals, flattened)

MathOverflow, If p_n is the largest prime factor of p_{n-1}+p_{n-2}+m, then p_n is bounded

Example

T(1,2) = 43 because the sequence in this case starts 2,3,3,7,11,19,31,17,7, and then repeats 5,13,19,11,31,43,5,7,13,7,7 in a cycle.
Array starts
5   43   5  43  43  13 137  43 151  29 ...
43  43  43  43  41  43  43  67  43  73 ...
5   43 131  43 131  43  41 131 137 137 ...
7   41  43  43  43  43 137  43 131  67 ...
43  43  41  43 131 131 131  43 131 151 ...
13  43 131  43  41  43  43  43  73  73 ...
17  43 137  43 151  47  43  41  41 131 ...
43  43 131  41  43  41  43  41  67 137 ...
23  43 137 131  43 151 137 137 197 137 ...
29  41  43 137  73  43 131  41 131 389 ...

Maple

M:= 20: # to get the first M antidiagonals
with(queue):
backprop:= proc(r, p)
  local t; global F;
  for t in Parents[r] do
    if F[t] < p then
      F[t]:= p;
      procname(t, p);
    fi
  od
end proc:
Verts:= {seq(seq([ithprime(i), ithprime(j)], i=1..M), j=1..M)}:
for v in Verts do F[v]:= max(v); Parents[v]:= {} od:
Agenda:= new(op(Verts)):
while not empty(Agenda) do
  t:= dequeue(Agenda);
  r:= [t[2], max(numtheory:-factorset(t[1]+t[2]+1))];
  if member(r, Verts) then
    Parents[r]:= Parents[r] union {t};
  else
    Verts:= Verts union {r};
    Parents[r]:= {t};
    enqueue(Agenda, r);
    F[r]:= max(r);
  fi;
  backprop(r, F[r]);
od:
seq(seq(F[[ithprime(m-j), ithprime(j)]], j=1..m-1), m=2..M+1);

Crossrefs

Cf. A006530.

Sequence in context: A132487 A178614 A360942 * A067927 A038546 A022891

Adjacent sequences: A284167 A284168 A284169 * A284171 A284172 A284173

Keyword

nonn,tabl

Author

Robert Israel, Mar 21 2017

Status

approved