

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).


1



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

1,1


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

Table of n, a(n) for n=1..23.
J. F. Alm and T. Herald, A Note on Prime Fibonacci Sequences, Fibonacci Quarterly 54:1 (2016), pp. 5558. arXiv:1507.04807 [math.NT], 2015.


PROG

(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)
(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


CROSSREFS

Cf. A214674.
Sequence in context: A121573 A196407 A156030 * A130140 A051417 A090368
Adjacent sequences: A255559 A255560 A255561 * A255563 A255564 A255565


KEYWORD

nonn


AUTHOR

Jeremy F. Alm, Jul 10 2015


EXTENSIONS

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


STATUS

approved



