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A347904
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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.
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2
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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, 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, 0, 0, 0, 19, 0, 17, 0, 0, 0, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13, 13
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OFFSET
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1,1
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COMMENTS
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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).
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LINKS
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FORMULA
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T(m,n) = 0 if m and n have a common factor.
T(m,n) = T(n,m+n) if m+n is not prime, otherwise T(m,n) = m+n.
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EXAMPLE
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Array begins:
m\n| 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16
---+---------------------------------------------------
1 | 2 3 7 5 11 7 23 17 19 11 23 13 41 29 31 17
2 | 3 0 5 0 7 0 41 0 11 0 13 0 43 0 17 0
3 | 5 5 0 7 13 0 17 11 0 13 103 0 29 17 0 19
4 | 5 0 7 0 23 0 11 0 13 0 41 0 17 0 19 0
5 | 7 7 11 13 0 11 19 13 23 0 43 17 31 19 0 37
6 | 7 0 0 0 11 0 13 0 0 0 17 0 19 0 0 0
7 | 17 11 13 11 17 13 0 23 41 17 29 19 53 0 37 23
8 | 19 0 11 0 13 0 37 0 17 0 19 0 89 0 23 0
9 | 11 11 0 13 19 0 23 17 0 19 31 0 149 23 0 41
10 | 11 0 13 0 0 0 17 0 19 0 53 0 23 0 0 0
11 | 13 13 17 19 37 17 43 19 29 31 0 23 37 103 41 43
12 | 13 0 0 0 17 0 19 0 0 0 23 0 101 0 0 0
13 | 29 17 19 17 23 19 47 29 31 23 59 37 0 41 43 29
14 | 31 0 17 0 19 0 0 0 23 0 61 0 67 0 29 0
15 | 17 17 0 19 0 0 29 23 0 0 37 0 41 29 0 31
16 | 17 0 19 0 47 0 23 0 59 0 103 0 29 0 31 0
T(2,7) = 41, because the first prime in A022113, excluding the two initial terms, is 41.
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PROG
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(Python)
# 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.
from sympy import isprime, gcd
if gcd(m, n) != 1:
return 0
m, n = n, m+n
while not isprime(n):
m, n = n, m+n
return n
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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