A214674 -id:A214674 - cp's OEIS frontend

A214892 Conway's subprime Fibonacci sequence starting with (4,1).

4, 1, 5, 3, 4, 7, 11, 9, 10, 19, 29, 24, 53, 11, 32, 43, 25, 34, 59, 31, 45, 38, 83, 11, 47, 29, 38, 67, 35, 51, 43, 47, 45, 46, 13, 59, 36, 19, 11, 15, 13, 14, 9, 23, 16, 13, 29, 21, 25, 23, 24, 47, 71, 59, 65, 62, 127, 63, 95, 79, 87, 83, 85, 84, 13, 97, 55, 76, 131, 69, 100, 13, 113

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (4, 1), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 136 after 8 terms on reaching (11, 9).

Wouter Meeussen, Table of n, a(n) for n = 1..144
Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099 [math.NT], 2012-2014.

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

(* see A214674 *)
a[1] = 4; a[2] = 1; a[n_] := a[n] = If[an = a[n-2]+a[n-1]; PrimeQ[an], an, an/FactorInteger[an][[1, 1]]]; Array[a, 80] (* Jean-François Alcover, Nov 17 2018 *)

A214898 Conway's subprime Fibonacci sequence, largest loop elements.

Original entry on oeis.org

2, 827, 607, 239, 191, 5693, 347

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with a pair of positive integers, but each new nonprime term is divided by its least prime factor. Recursion enters a loop of length A214897(n), of which the largest element a(n) is prime (this sequence).

Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

A272636 a(0)=0, a(1)=1; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7, 1, 2, 3, 5, 2, 7

Offset: 0

N. J. A. Sloane, May 05 2016

nonn

Periodic with period {1,2,3,5,2,7}.
James Propp, in a posting to the Math Fun list, asks if every sequence of positive numbers satisfying the same recurrence will eventually merge with this sequence (as A272638 does). The answer is no, Fred W. Helenius found infinitely many counterexamples, including A272637. See A272639 for other counterexamples which start 1,x.
Other counterexamples found by Helenius include [n, 2n, 3n, 5n, 2n, 7n] (period 6) where n is any squarefree positive integer coprime to 210 = 2*3*5*7.

Index entries for linear recurrences with constant coefficients, signature (0, 0, 0, 0, 0, 1).

Cf. A007913 (squarefree part of n), A000045, A272637, A272638, A272639.
See A165911 for a similar sequence.
See also A214674, A214892-A214898.

{0, 1}~Join~LinearRecurrence[{0, 0, 0, 0, 0, 1}, {1, 2, 3, 5, 2, 7}, 120] (* Jean-François Alcover, Nov 16 2019 *)

from sympy.ntheory.factor_ import core
l=[0, 1]
for n in range(2, 101):
    l.append(core(l[n - 1] + l[n - 2]))
print(l) # Indranil Ghosh, Jun 03 2017

A282813 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the starting index of the first loop in Conway's subprime Fibonacci sequence with starting values of n and k.

Original entry on oeis.org

38, 47, 37, 38, 1, 46, 7, 45, 36, 7, 47, 37, 1, 34, 6, 60, 38, 33, 4, 44, 34, 8, 35, 5, 1, 35, 8, 59, 47, 37, 2, 34, 6, 3, 37, 79, 7, 45, 36, 7, 1, 73, 32, 103, 45, 24, 44, 78, 58, 72, 30, 3, 3, 100, 78, 35, 9, 4, 74, 31, 1, 43, 35, 74, 35, 23, 55, 108, 35, 98

Offset: 1

Peter Kagey, Feb 21 2017

			T(1, 1) = 38 because 38 is the starting index of the first loop in Conway's subprime Fibonacci sequence when starting with 1, 1 (A214674). (i.e. A214674(n) = A214674(n + k) for n >= 38 = T(1, 1) and k = A282814(1,1).)
Upper-left corner of table:
   38  37  46   7   6  34  59  79  45  78 ...
   47   1  36  34  44   8  37 103 100  35 ...
   38  45   1   4  35   3  32   3  74  92 ...
    7  37  33   1   6  73   3  35  50  98 ...
   47  38   5  34   1  30  43  55 105   3 ...
   60  35   2   7  72   1  33  33  45  22 ...
    8  37  36  58  31  42   1 103  75  52 ...
   47  45  78  74 102  34  32   1  20  61 ...
    7  44   4  98  44  73 104  74   1   1 ...
   24   9  35   4   2  34  97  21  60   1 ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A214674, A282812, A282814.

A282814 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the length of the loop in Conway's subprime Fibonacci sequence with starting values of n and k.

Original entry on oeis.org

18, 18, 18, 18, 1, 18, 136, 18, 18, 136, 18, 18, 1, 18, 136, 18, 18, 18, 136, 18, 18, 136, 18, 136, 1, 18, 136, 18, 18, 18, 1, 18, 136, 1, 18, 18, 136, 18, 18, 136, 1, 18, 18, 136, 18, 18, 18, 18, 18, 18, 18, 136, 136, 136, 18, 18, 136, 1, 18, 18, 1, 18, 18

Offset: 1

Peter Kagey, Feb 21 2017

All terms are in A214897, which is conjectured to be finite. - Peter Kagey, Nov 04 2017

			T(1, 1) = 18 because 18 is the loop length in Conway's subprime Fibonacci sequence when starting with 1, 1 (A214674). (i.e., A214674(n) = A214674(n + 18) for n >= 38 = A282813(1, 1).)
Upper-left corner of table:
   18  18  18 136 136  18  18  18  18  18 ...
   18   1  18  18  18 136  18 136 136  18 ...
   18  18   1 136  18   1  18 136  18  18 ...
  136  18  18   1 136  18 136  18  18 136 ...
   18  18 136  18   1  18  18  18 136   1 ...
   18  18   1 136  18   1  18  18  18  18 ...
  136  18  18  18  18  18   1 136  18  18 ...
   18  18  18  18 136  18  18   1  18  18 ...
  136  18   1  18  18  18 136  18   1 136 ...
   18 136  18 136   1  18 136  18  18   1 ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A214674, A214897, A282813, A282814.

A165911 a(n) = squarefree kernel (or radical) of a(n-1) + a(n-2), with a(0)=0 and a(1)=1.

Original entry on oeis.org

0, 1, 1, 2, 3, 5, 2, 7, 3, 10, 13, 23, 6, 29, 35, 2, 37, 39, 38, 77, 115, 6, 11, 17, 14, 31, 15, 46, 61, 107, 42, 149, 191, 170, 19, 21, 10, 31, 41, 6, 47, 53, 10, 21, 31, 26, 57, 83, 70, 51, 11, 62, 73, 15, 22, 37, 59, 6, 65, 71, 34, 105, 139, 122, 87, 209, 74, 283, 357, 10

Offset: 0

Franklin T. Adams-Watters, Sep 30 2009

nonn

The squarefree kernel (or radical) of n is the largest squarefree divisor of n, A007947.
Through n=1688, this sequence does not loop. Does it grow indefinitely, or is it eventually periodic?
The graph suggests that the sequence had a chance to go into a cycle between terms 100 and 150, but by the time we get to 1688 terms the sequence seems to have reached escape velocity and there is no further hope of this happening. (Of course this is not a rigorous argument.) - N. J. A. Sloane, May 06 2016
If we take the squarefree part (A007913) instead of the squarefree kernel, the sequence is periodic from n=1, repeating 1,2,3,5,2,7. See A272636.
From Fred W. Helenius, May 07 2016: (Start)
There are many examples of initial conditions for this recurrence that produce sequences that cycle.
Examples that arise where a(0) and a(1) are not coprime:
  - 2, 2 (period 1)
  - 3, 3, 6, 3, 3, 6 (period 3)
  - 5, 10, 15, 5, 10, 15 (period 3)
Examples of periodic sequences starting with coprime initial values:
  - 15 146 161 307 78 385 463 106 569 (period 9)
  - 222 1589 1811 170 1981 717 2698 3415 6113 2382 8495 10877 9686 20563 10083 30646 3133 33779 4614 38393 43007 4070 47077 17049 64126 16235 26787 6146 32933 39079 36006 75085 111091 5818 5083 10901 (period 46)
  - 770 559 1329 118 1447 1565 1506 3071 4577 478 5055 5533 5294 1203 6497 (period 15)
(End)

Franklin T. Adams-Watters and N. J. A. Sloane, Table of n, a(n) for n = 0..1688, May 07 2016 [First 1000 terms from Franklin T. Adams-Watters]

Cf. A007947.

See A000045, A272636, A272637, A272638, A272639 for similar sequences. See also A214674, A214892-A214898.

nxt[{a_,b_}]:={b,Select[Divisors[a+b],SquareFreeQ][[-1]]}; NestList[nxt,{0,1},70][[All,1]] (* Harvey P. Dale, Jul 31 2018 *)

rad(n)=local(fm);fm=factor(n);prod(k=1,matsize(fm)[1],fm[k,1])
v=vector(100,n,1);for(n=3,100,v[n]=rad(v[n-1]+v[n-2]))

from operator import mul
from sympy import primefactors
from functools import reduce
def rad(n): return 1 if n<2 else reduce(mul, primefactors(n))
l=[0, 1]
for n in range(2, 101):
    l.append(rad(l[n - 1] + l[n - 2]))
print(l) # Indranil Ghosh, Jun 03 2017

A282812 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the largest value in Conway's subprime Fibonacci sequence with starting values of n and k.

Original entry on oeis.org

239, 239, 239, 239, 2, 239, 347, 239, 239, 347, 239, 239, 3, 239, 347, 239, 239, 239, 347, 239, 239, 347, 239, 347, 4, 239, 347, 239, 239, 239, 6, 239, 347, 6, 239, 239, 347, 239, 239, 347, 5, 239, 239, 463, 239, 97, 239, 239, 239, 239, 239, 347, 347, 463

Offset: 1

Peter Kagey, Feb 21 2017

			T(1, 1) = 239 because 239 is the largest value in Conway's subprime Fibonacci sequence when starting with 1, 1 (A214674).
Upper-left corner of table:
  239 239 239 347 347 239 239 239 239 239 ...
  239   2 239 239 239 347 239 463 463 239 ...
  239 239   3 347 239   6 239 347 239 239 ...
  347 239 239   4 347 239 347 239 239 463 ...
  239 239 347 239   5 239 239 239 463  10 ...
  239 239   6 347 239   6 239 239 239 109 ...
  347 239 239 239 239 239   7 463 239 239 ...
  239 239 239 239 463 239 239   8 109 239 ...
  347 239   9 733 239 239 463 239   9 347 ...
   97 347 239 347  10 239 463 109 239  10 ...
  ...

Peter Kagey, Table of n, a(n) for n = 1..10000

Cf. A214674, A282813, A282814.

A214897 Conway's subprime Fibonacci sequence: cycle lengths.

Original entry on oeis.org

1, 10, 11, 18, 19, 56, 136

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with a pair of positive integers, but each new nonprime term is divided by its least prime factor. The recursion enters a loop of length a(n) after a finite number of steps. Conjecture: the list of loops is complete (checked to [10^5, 10^5]), loops of length a(n) are unique and no infinite chains exist.

Richard K. Guy, Tanya Khovanova and Julian Salazar, Conway's subprime Fibonacci sequences, arXiv:1207.5099v1 [math.NT]

Cf. A000045, A020639, A175723, A214551, A014682, A214674, A214892-A214898.

Mathematica
```
see A214674
```

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).

Original entry on oeis.org

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

Jeremy F. Alm, Jul 10 2015

nonn

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.

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.

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

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

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))

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

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

A117339 a(n) = a(n-1) + a(n-2); if a(n) is not prime divide a(n) by its largest prime factor.

Original entry on oeis.org

1, 1, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2, 3, 5, 4, 3, 7, 2

Offset: 1

Zak Seidov, Mar 09 2006

Sequence repeats cyclically. Conjecture: sequence ends with cycle for any positive a(1) and a(2). This has been checked for a(1)=1 and a(2)=1,...,2000. Some other cases shown below: a(1), a(2), length of s, sequence s 1,1,10,{1,1,2,3,5,4,3,7,2,3}, 1,2,9,{1,2,3,5,4,3,7,2,3}, 1,3,9,{1,3,2,5,7,4,11,3,2}, 1,4,20,{1,4,5,3,4,7,11,6,17,23,8,31,3,2,5,7,4,11,3,2}, 1,5,12,{1,5,2,7,3,2,5,7,4,11,3,2}, 1,6,23,{1,6,7,13,4,17,3,4,7,11,6,17,23,8,31,3,2,5,7,4,11,3,2}, 1,7,9,{1,7,4,11,3,2,5,7,4}, 1,8,30,{1,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}, 1,9,34,{1,9,2,11,13,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}, 1,10,10,{1,10,11,3,2,5,7,4,11,3}. Even exotic seeds give no long sequences, e.g. 967,3000,24,{967,3000,3967,6967,154,7121,75,28,103,131,18,149,167,4,9,13,2,3,5,4,3,7,2,3}. For a(1)=1, maximal length = 55 is found for a(2)=1207: 1,1207,55,{1,1207,8,243,251,26,277,3,40,43,83,18,101,7,36,43,79,2,27,29,8,37,9,2,11,13,8,3,11,2,13,3,8,11,19,6,5,11,8,19,9,4,13,17,6,23,29,4,3,7,2,3,5,4,3}.

Index entries for linear recurrences with constant coefficients, signature (0,0,0,0,0,1).

Cf. A214674.

G.f.: -(6*x^8+2*x^7+4*x^6+5*x^5+3*x^4+2*x^3+x^2+x)/(x^6-1). - Alois P. Heinz, Apr 20 2023

cp's OEIS Frontend

A214892 Conway's subprime Fibonacci sequence starting with (4,1).

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A214898 Conway's subprime Fibonacci sequence, largest loop elements.

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A272636 a(0)=0, a(1)=1; thereafter a(n) = squarefree part of a(n-1)+a(n-2).

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A282813 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the starting index of the first loop in Conway's subprime Fibonacci sequence with starting values of n and k.

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A282814 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the length of the loop in Conway's subprime Fibonacci sequence with starting values of n and k.

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A165911 a(n) = squarefree kernel (or radical) of a(n-1) + a(n-2), with a(0)=0 and a(1)=1.

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A282812 Table read by antidiagonals (n > 0, k > 0): T(n, k) is the largest value in Conway's subprime Fibonacci sequence with starting values of n and k.

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A214897 Conway's subprime Fibonacci sequence: cycle lengths.

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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).

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