A255562 A reversed prime Fibonacci sequence: a(n+2) is the smallest odd prime such that a(n) is the smallest odd prime divisor of a(n+1)+a(n+2).

3, 5, 7, 3, 11, 7, 37, 19, 277, 331, 223, 439, 7, 406507, 67, 330515394367, 967, 10576492618777, 116041, 223724392248491824062507397, 3691561, 100105207373914057144918297314160710207525630111509317, 423951181

Offset: 1

Author

Jeremy F. Alm, Jul 10 2015

Keywords

• nonn

Comments

The sequence satisfies a(1) = 3, a(2) = 5, and a(n+2) is the smallest odd prime with the following property: a(n) is the smallest odd prime divisor of a(n+1)+a(n+2). It is a provably infinite sequence. It is also the "reverse" of a prime Fibonacci sequence terminating in 5,3. A prime Fibonacci sequence satisfies the following relation: a(n+2) is the smallest odd prime dividing a(n)+a(n+1), unless a(n)+a(n+1) is a power of two, in which case the sequence terminates. Prime Fibonacci sequences provably terminate, but provably can be extended indefinitely to the left.

Links

• J. F. Alm and T. Herald, A Note on Prime Fibonacci Sequences, Fibonacci Quarterly 54:1 (2016), pp. 55-58. arXiv:1507.04807 [math.NT], 2015.

Crossrefs

Cf. A214674, A352955 (starting with 11,19).

Programs

PARI

lista(nn) = {print1(pp=3, ", "); print1(p=5, ", "); for (n=1, nn, forprime(q=3, , s = (p+q)/ 2^(valuation(p+q, 2)); if ((s!=1) && pp == factor(s)[1,1], np = q; break);); print1(np, ", "); pp = p; p = np;);} \\ Michel Marcus, Jul 11 2015

Python

import math
def sieve(n):
    r = int(math.floor(math.sqrt(n)))
    composites = [j for i in range(2,r+1) for j in range(2*i, n, i)]
    primes = set(range(2,n)).difference(set(composites))
    return sorted(primes)
Primes = sieve(1000000)
Odd_primes = Primes[1:]
def find_smallest_odd_div(n):
    for p in Odd_primes:
        if n % p == 0:
            return p
def next_term(a,b):
    for p in Odd_primes:
        if (p + b) % a == 0:
            if find_smallest_odd_div(p+b) == a:
                return p
def compute_reversed_seq(a,b):
    seq = [a,b]
    while seq[-1] != None:
        seq.append(next_term(seq[-2],seq[-1]))
    return seq[:len(seq)-1]
print(compute_reversed_seq(3,5))

Python

from sympy import isprime, factorint
from itertools import islice
def rem2(n):
    while n%2 == 0: n //= 2
    return n
def agen():
    b, c = 3, 5
    yield 3
    while True:
        yield c
        k = (c+2)//b + 1
        m = b*k
        while not isprime(m-c) or min(factorint(rem2(k)), default=b+1) < b:
            m += b
            k += 1
        b, c = c, m-c
print(list(islice(agen(), 19))) # Michael S. Branicky, Apr 12 2022

Extensions

a(16)-a(23) from Giovanni Resta, Jul 17 2015