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%I #57 Nov 14 2023 12:41:59
%S 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,
%T 135,149,142,97,239,168,37,41,39,40,79,17,48,13,61,37,49,43,46,89,45,
%U 67,56,41,97,69,83,76,53,43,48,13
%N Conway's subprime Fibonacci sequence.
%C 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).
%D Siobhan Roberts, Genius At Play: The Curious Mind of John Horton Conway, Bloomsbury, 2015, pages xx-xxi.
%H Peter Kagey, <a href="/A214674/b214674.txt">Table of n, a(n) for n = 1..250</a>
%H Sara Barrows, Emily Noye, Sarah Uttormark, and Matthew Wright, <a href="https://doi.org/10.1080/07468342.2023.2263109">Three's A Crowd: An Exploration of Subprime Tribonacci Sequences</a>, College Math. J. (2023).
%H Richard K. Guy, Tanya Khovanova and Julian Salazar, <a href="http://arxiv.org/abs/1207.5099">Conway's subprime Fibonacci sequences</a>, arXiv:1207.5099 [math.NT], 2012-2014.
%H Tanya Khovanova, <a href="http://blog.tanyakhovanova.com/2012/07/conways-subprime-fibonacci-sequences/">Conway’s Subprime Fibonacci Sequences</a>, Math Blog, July 2012.
%H <a href="/index/Rec#order_18">Index entries for linear recurrences with constant coefficients</a>, signature (0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1).
%t 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}]
%t 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 *)
%o (PARI) 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}
%o fatw(99) /* _M. F. Hasler_, Jul 25 2012 */
%Y Cf. A000045, A020639, A117339, A175723, A214551, A014682, etc.
%Y Cf. A214892-A214898, A282812, A282813, A282814.
%Y See also A165911, A272636, A255562, etc.
%K nonn,easy
%O 1,3
%A _Wouter Meeussen_, Jul 25 2012
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