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

1,3

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

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).

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

Cf. A000045, A020639, A117339, A175723, A214551, A014682, etc.

Cf. A214892-A214898, A282812, A282813, A282814.

See also A165911, A272636, A255562, etc.

Sequence in context: A117339 A096016 A123274 * A185332 A023818 A102149

Adjacent sequences: A214671 A214672 A214673 * A214675 A214676 A214677

Keyword

nonn,easy

Author

Wouter Meeussen, Jul 25 2012

Status

approved