login
A214674
Conway's subprime Fibonacci sequence.
15
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
1,3
COMMENTS
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).
REFERENCES
Siobhan Roberts, Genius At Play: The Curious Mind of John Horton Conway, Bloomsbury, 2015, pages xx-xxi.
LINKS
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).
MATHEMATICA
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 *)
PROG
(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}
fatw(99) /* M. F. Hasler, Jul 25 2012 */
CROSSREFS
KEYWORD
nonn,easy
AUTHOR
Wouter Meeussen, Jul 25 2012
STATUS
approved