A347905 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.

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, 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, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 4, 0, 6, 0, 3, 0, 0, 0, 3, 0, 3, 0, 4

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 largest value of T(m,n) for m, n <= 5000 is T(1591,300) = 17262.

Links

Table of n, a(n) for n=1..91.

Formula

T(m,n) = 0 if m and n have a common factor.

T(m,n) = T(n,m+n) + 1 if m+n is not prime, otherwise T(m,n) = 2.

Example

Array begins:
  m\n|  1  2  3  4  5  6  7  8  9 10 11 12 13 14 15 16 17 18 19 20
  ---+------------------------------------------------------------
   1 |  2  2  3  2  3  2  4  3  3  2  3  2  4  3  3  2  4  2  4  3
   2 |  2  0  2  0  2  0  5  0  2  0  2  0  4  0  2  0  2  0  4  0
   3 |  3  2  0  2  3  0  3  2  0  2  6  0  3  2  0  2  3  0  3  2
   4 |  2  0  2  0  4  0  2  0  2  0  4  0  2  0  2  0  4  0  2  0
   5 |  3  2  3  3  0  2  3  2  3  0  4  2  3  2  0  3  4  2  3  0
   6 |  2  0  0  0  2  0  2  0  0  0  2  0  2  0  0  0  2  0  5  0
   7 |  4  3  3  2  3  2  0  3  4  2  3  2  4  0  3  2  3  3  4  3
   8 |  4  0  2  0  2  0  4  0  2  0  2  0  5  0  2  0  4  0  4  0
   9 |  3  2  0  2  3  0  3  2  0  2  3  0  6  2  0  3  3  0  3  2
  10 |  2  0  2  0  0  0  2  0  2  0  4  0  2  0  0  0  4  0  2  0
  11 |  3  2  3  3  4  2  4  2  3  3  0  2  3  5  3  3  4  2  4  2
  12 |  2  0  0  0  2  0  2  0  0  0  2  0  5  0  0  0  2  0  2  0
  13 |  4  3  3  2  3  2  4  3  3  2  4  3  0  3  3  2  3  2  4  3
  14 |  4  0  2  0  2  0  0  0  2  0  4  0  4  0  2  0  2  0  5  0
  15 |  3  2  0  2  0  0  3  2  0  0  3  0  3  2  0  2  6  0  3  0
  16 |  2  0  2  0  4  0  2  0  4  0  5  0  2  0  2  0  4  0  4  0
  17 |  3  2  3  5 10  2  3  6  4  3  4  2  3  2  3  5  0  3  7  2
  18 |  2  0  0  0  2  0  5  0  0  0  2  0  2  0  0  0  5  0  2  0
  19 |  4  3  4  2  3  3  4  5  3  2  3  2  6  3  4  5  3  2  0  3
  20 |  4  0  2  0  0  0  4  0  2  0  2  0  4  0  0  0  2  0  4  0
T(2,7) = 5, because 5 is the smallest k >= 2 for which A022113(k) is prime.

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 A347905(m, n):
    if gcd(m, n) != 1:
        return 0
    m, n = n, m+n
    k=2
    while not isprime(n):
        m, n = n, m+n
        k += 1
    return k

Crossrefs

Cf. A022113, A082411, A083104, A083105, A083216, A221286, A347904.

Sequence in context: A004489 A305384 A112599 * A308119 A307299 A307298

Adjacent sequences: A347902 A347903 A347904 * A347906 A347907 A347908

Keyword

nonn,tabl

Author

Pontus von Brömssen, Sep 18 2021

Status

approved