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

1,1

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).

Links

Table of n, a(n) for n=1..78.

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.

Example

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.

Prog

(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

Crossrefs

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

Sequence in context: A028257 A100228 A111003 * A289277 A140182 A340753

Adjacent sequences: A347901 A347902 A347903 * A347905 A347906 A347907

Keyword

nonn,tabl

Author

Pontus von Brömssen, Sep 18 2021

Status

approved