A214892 -id:A214892 - 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 */

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

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

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
```

A214893 Conway's subprime Fibonacci sequence starting with (18, 5).

Original entry on oeis.org

18, 5, 23, 14, 37, 17, 27, 22, 7, 29, 18, 47, 13, 30, 43, 73, 58, 131, 63, 97, 80, 59, 139, 99, 119, 109, 114, 223, 337, 280, 617, 299, 458, 757, 405, 581, 493, 537, 515, 526, 347, 291, 319, 305, 312, 617, 929

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (18, 5), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 56 after 26 terms on reaching (119, 109).

Wouter Meeussen, Table of n, a(n) for n = 1..82
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.

see A214674
nxt[{a_,b_}]:=Module[{c=a+b},{b,If[PrimeQ[c],c,c/FactorInteger[c][[1,1]]]}]; Transpose[NestList[nxt,{18,5},82]][[1]] (* Harvey P. Dale, Oct 19 2012 *)

A214894 Conway's subprime Fibonacci sequence starting with (10, 18).

Original entry on oeis.org

10, 18, 14, 16, 15, 31, 23, 27, 25, 26, 17, 43, 30, 73, 103, 88, 191, 93, 142, 47, 63, 55, 59, 57, 58, 23, 27

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (10, 18), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 19 after 8 terms on reaching (23, 27).

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

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

Mathematica
```
see A214674
```

A214895 Conway's subprime Fibonacci sequence starting with (23, 162).

Original entry on oeis.org

23, 162, 37, 199, 118, 317, 145, 231, 188, 419, 607, 513, 560, 37, 199

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (23, 162), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 11 after 4 terms on reaching (37, 199).

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

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

Mathematica
```
see A214674
```

A214896 Conway's subprime Fibonacci sequence starting with (382, 127).

Original entry on oeis.org

382, 127, 509, 318, 827, 229, 528, 757, 257, 507, 382, 127

Offset: 1

Wouter Meeussen, Jul 29 2012

nonn

Similar to the Fibonacci recursion starting with (382, 127), but each new nonprime term is divided by its least prime factor. Sequence enters a loop of length 10 after 2 terms on reaching (382, 127).

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
```

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A214674 Conway's subprime Fibonacci sequence.

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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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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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A214897 Conway's subprime Fibonacci sequence: cycle lengths.

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A214893 Conway's subprime Fibonacci sequence starting with (18, 5).

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A214894 Conway's subprime Fibonacci sequence starting with (10, 18).

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A214895 Conway's subprime Fibonacci sequence starting with (23, 162).

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A214896 Conway's subprime Fibonacci sequence starting with (382, 127).

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