A255562 -id:A255562 - cp's OEIS frontend

A214674 Conway's subprime Fibonacci sequence.

1, 1, 2, 3, 5, 4, 3, 7, 5, 6, 11, 17, 14, 31, 15, 23, 19, 21, 20, 41, 61, 51, 56, 107, 163, 135, 149, 142, 97, 239, 168, 37, 41, 39, 40, 79, 17, 48, 13, 61, 37, 49, 43, 46, 89, 45, 67, 56, 41, 97, 69, 83, 76, 53, 43, 48, 13

Offset: 1

Wouter Meeussen, Jul 25 2012

Similar to the Fibonacci recursion starting with (1, 1), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 18 after 38 terms on reaching (48, 13).

Siobhan Roberts, Genius At Play: The Curious Mind of John Horton Conway, Bloomsbury, 2015, pages xx-xxi.

Peter Kagey, Table of n, a(n) for n = 1..250
Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright, Three's A Crowd: An Exploration of Subprime Tribonacci Sequences, College Math. J. (2023).
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.
Tanya Khovanova, Conway’s Subprime Fibonacci Sequences, Math Blog, July 2012.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).

Cf. A000045, A020639, A117339, A175723, A214551, A014682, etc.
Cf. A214892-A214898, A282812, A282813, A282814.
See also A165911, A272636, A255562, etc.

guyKhoSal[{a_, b_}] := Block[{c, l, r}, c = NestWhile[(p = Tr[Take[#, -2]]; If[PrimeQ[p], q = p, q = p/Part[FactorInteger[p, FactorComplete -> False], 1, 1]]; Flatten[{#, q}]) &, {a, b}, FreeQ[Partition[#1, 2, 1], Take[#2, -2]] &, 2, 1000]; l = Length[c]; r = Tr@Position[Partition[c,2,1], Take[c,-2], 1, 1]; l-r-1; c]; guyKhoSal[{1,1}]
f[s_List] := Block[{a = s[[-2]] + s[[-1]]}, If[ PrimeQ[a], Append[s, a], Append[s, a/FactorInteger[a][[1, 1]] ]]]; Nest[f, {1, 1}, 73] (* Robert G. Wilson v, Aug 09 2012 *)

fatw(n,a=[0,1],p=[])={for(i=2,n,my(f=factor(a[i]+a[i-1])~);for(k=1,#f,setsearch(p,f[1,k])&next;f[2,k]--;p=setunion(p,Set(f[1,k]));break);a=concat(a,factorback(f~)));a}
fatw(99) /* M. F. Hasler, Jul 25 2012 */

A352955 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), starting with a(1) = 11 and a(2) = 19.

Original entry on oeis.org

11, 19, 3, 73, 5, 1163, 17, 2309, 3, 147773, 7, 2364361, 43, 75659509, 109, 605275963, 601, 732718084457515921, 2767, 1500606636968992603441, 145687, 192077649532031053094761, 10273, 103120902879077884687304760481759, 5036623

Offset: 1

Michel Marcus, Apr 11 2022

nonn

a(26) > 4.34448*10^41. - Michael S. Branicky, Apr 12 2022

Jeremy F. Alm and Taylor Herald, A Note on Prime Fibonacci Sequences, Fibonacci Quart. 54 (2016), no. 1, 55-58.

Cf. A255562 (starting with 3,5).

from sympy import isprime, factorint
from itertools import islice
def rem2(n):
    while n%2 == 0: n //= 2
    return n
def agen():
    b, c = 11, 19
    yield 11
    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(), 17))) # Michael S. Branicky, Apr 12 2022

a(18)-a(25) from Michael S. Branicky, Apr 11 2022

cp's OEIS Frontend

A214674 Conway's subprime Fibonacci sequence.

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