A165911 -id:A165911 - 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 */

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

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A214674 Conway's subprime Fibonacci sequence.

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