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%I #9 Sep 20 2021 11:49:30
%S 2,3,3,7,0,5,5,5,5,5,11,0,0,0,7,7,7,7,7,7,7,23,0,13,0,11,0,17,17,41,0,
%T 23,13,0,11,19,19,0,17,0,0,0,13,0,11,11,11,11,11,11,11,11,11,11,11,23,
%U 0,0,0,19,0,17,0,0,0,13,13,13,13,13,13,13,13,13,13,13,13,13
%N Array read by antidiagonals, m, n >= 1: T(m,n) is 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. The smallest (in the sense that m+n is as small as possible) known case where this occurs appears to be m = 106276436867, n = 35256392432 (Vsemirnov's sequence, A221286).
%F T(m,n) = 0 if m and n have a common factor.
%F T(m,n) = T(n,m+n) if m+n is not prime, otherwise T(m,n) = m+n.
%e Array begins:
%e m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
%e ---+---------------------------------------------------
%e 1 | 2 3 7 5 11 7 23 17 19 11 23 13 41 29 31 17
%e 2 | 3 0 5 0 7 0 41 0 11 0 13 0 43 0 17 0
%e 3 | 5 5 0 7 13 0 17 11 0 13 103 0 29 17 0 19
%e 4 | 5 0 7 0 23 0 11 0 13 0 41 0 17 0 19 0
%e 5 | 7 7 11 13 0 11 19 13 23 0 43 17 31 19 0 37
%e 6 | 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0
%e 7 | 17 11 13 11 17 13 0 23 41 17 29 19 53 0 37 23
%e 8 | 19 0 11 0 13 0 37 0 17 0 19 0 89 0 23 0
%e 9 | 11 11 0 13 19 0 23 17 0 19 31 0 149 23 0 41
%e 10 | 11 0 13 0 0 0 17 0 19 0 53 0 23 0 0 0
%e 11 | 13 13 17 19 37 17 43 19 29 31 0 23 37 103 41 43
%e 12 | 13 0 0 0 17 0 19 0 0 0 23 0 101 0 0 0
%e 13 | 29 17 19 17 23 19 47 29 31 23 59 37 0 41 43 29
%e 14 | 31 0 17 0 19 0 0 0 23 0 61 0 67 0 29 0
%e 15 | 17 17 0 19 0 0 29 23 0 0 37 0 41 29 0 31
%e 16 | 17 0 19 0 47 0 23 0 59 0 103 0 29 0 31 0
%e T(2,7) = 41, because the first prime in A022113, excluding the two initial terms, is 41.
%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 A347904(m,n):
%o if gcd(m,n) != 1:
%o return 0
%o m,n = n,m+n
%o while not isprime(n):
%o m,n = n,m+n
%o return n
%Y Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347905.
%K nonn,tabl
%O 1,1
%A _Pontus von Brömssen_, Sep 18 2021