%I #10 Sep 20 2021 11:50:03
%S 2,2,2,3,0,3,2,2,2,2,3,0,0,0,3,2,2,2,2,2,2,4,0,3,0,3,0,4,3,5,0,4,3,0,
%T 3,4,3,0,3,0,0,0,3,0,3,2,2,2,2,2,2,2,2,2,2,3,0,0,0,3,0,3,0,0,0,3,2,2,
%U 2,2,2,2,2,2,2,2,2,2,4,0,6,0,3,0,0,0,3,0,3,0,4
%N Array read by antidiagonals, m, n >= 1: T(m,n) is the position of the first prime (after the two initial terms) in the Fibonacci-like sequence with initial terms m and n, or 0 if no such prime exists.
%C There are cases where T(m,n) = 0 even when m and n are coprime; see A082411, A083104, A083105, A083216, and A221286.
%C The largest value of T(m,n) for m, n <= 5000 is T(1591,300) = 17262.
%F T(m,n) = 0 if m and n have a common factor.
%F T(m,n) = T(n,m+n) + 1 if m+n is not prime, otherwise T(m,n) = 2.
%e Array begins:
%e m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20
%e ---+------------------------------------------------------------
%e 1 | 2 2 3 2 3 2 4 3 3 2 3 2 4 3 3 2 4 2 4 3
%e 2 | 2 0 2 0 2 0 5 0 2 0 2 0 4 0 2 0 2 0 4 0
%e 3 | 3 2 0 2 3 0 3 2 0 2 6 0 3 2 0 2 3 0 3 2
%e 4 | 2 0 2 0 4 0 2 0 2 0 4 0 2 0 2 0 4 0 2 0
%e 5 | 3 2 3 3 0 2 3 2 3 0 4 2 3 2 0 3 4 2 3 0
%e 6 | 2 0 0 0 2 0 2 0 0 0 2 0 2 0 0 0 2 0 5 0
%e 7 | 4 3 3 2 3 2 0 3 4 2 3 2 4 0 3 2 3 3 4 3
%e 8 | 4 0 2 0 2 0 4 0 2 0 2 0 5 0 2 0 4 0 4 0
%e 9 | 3 2 0 2 3 0 3 2 0 2 3 0 6 2 0 3 3 0 3 2
%e 10 | 2 0 2 0 0 0 2 0 2 0 4 0 2 0 0 0 4 0 2 0
%e 11 | 3 2 3 3 4 2 4 2 3 3 0 2 3 5 3 3 4 2 4 2
%e 12 | 2 0 0 0 2 0 2 0 0 0 2 0 5 0 0 0 2 0 2 0
%e 13 | 4 3 3 2 3 2 4 3 3 2 4 3 0 3 3 2 3 2 4 3
%e 14 | 4 0 2 0 2 0 0 0 2 0 4 0 4 0 2 0 2 0 5 0
%e 15 | 3 2 0 2 0 0 3 2 0 0 3 0 3 2 0 2 6 0 3 0
%e 16 | 2 0 2 0 4 0 2 0 4 0 5 0 2 0 2 0 4 0 4 0
%e 17 | 3 2 3 5 10 2 3 6 4 3 4 2 3 2 3 5 0 3 7 2
%e 18 | 2 0 0 0 2 0 5 0 0 0 2 0 2 0 0 0 5 0 2 0
%e 19 | 4 3 4 2 3 3 4 5 3 2 3 2 6 3 4 5 3 2 0 3
%e 20 | 4 0 2 0 0 0 4 0 2 0 2 0 4 0 0 0 2 0 4 0
%e T(2,7) = 5, because 5 is the smallest k >= 2 for which A022113(k) is prime.
%o (Python)
%o # Note that in the (rare) case when m and n are coprime but there are no primes in the Fibonacci-like sequence, this function will go into an infinite loop.
%o from sympy import isprime,gcd
%o def A347905(m,n):
%o if gcd(m,n) != 1:
%o return 0
%o m,n = n,m+n
%o k=2
%o while not isprime(n):
%o m,n = n,m+n
%o k += 1
%o return k
%Y Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347904.
%K nonn,tabl
%O 1,1
%A _Pontus von Brömssen_, Sep 18 2021