A347904 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.

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

Offset: 1

Author

Pontus von Brömssen, Sep 18 2021

Keywords

• nonn
• tabl

Comments

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).

Examples

			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.
		

Crossrefs

Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347905.

Programs

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
def A347904(m,n):
    if gcd(m,n) != 1:
        return 0
    m,n = n,m+n
    while not isprime(n):
        m,n = n,m+n
    return n

Formula

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.