This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A224523 #37 May 15 2013 08:41:22
%S A224523 1,3,6,9,9,12,15,27,12,12,15,15,15,15,15,15,15,42,42,42,42,42,42,90,
%T A224523 72,36,36,36,36,36,36,36,36,36,54,54,66,102,102,102,102,102,102,102,
%U A224523 102,36,36,36,36,36,36,24,24,24,24,24,24,24,24,24,24,24,24,24
%N A224523 Let {f_n(k)} be the n-th sequence of Fibonacci-like numbers defined by recursion f_n(0) = 0, f_n(1) = 1 and, for k>=2, f_n(k) = f_n(k-1) + f_n(k-2) divided by maximal possible powers of primes >= prime(n). a(n) is length of the smallest period of {f_n(k)}, and a(n)=0, if {f_n(k)} is not eventually periodic.
%C A224523 Conjecture. For n>=1, a(n)>0.
%C A224523 If Conjecture is true, then we have an answer in affirmative on question in A078414.
%H A224523 Peter J. C. Moses, <a href="/A224523/b224523.txt">Table of n, a(n) for n = 1..4000</a>
%e A224523 Let n=9, prime(9)=23. Sequence {f_9(k)} begins 0,1,1,2,3,5,8,13,21,34,55. Now 34 + 55 = 89 is prime >=23, so the following terms are 89/89 = 1,56,57. Further, since 56 + 57 = 113 is prime >=23, then the following term is 113/113 = 1 and, since 57 + 1 = 58=29*2, then the following term is 58/29 = 2. Now we have period {1,2,3,5,8,13,21,34,55,1,56,57} with length 12. Thus a(9)=12.
%t A224523 seqPosition[{list_,seqtofind_}]:=Position[Partition[list,Length[#],1],Flatten[{___,#,___}]]&[seqtofind];Table[Clear[a];a[0]:=0;a[1]:=1;a[n_]:=a[n]=#/(Product[Prime[i]^IntegerExponent[#,Prime[i]],{i,z,PrimePi[#]+1}])&[(a[n-1]+a[n-2])];NestWhile[#+2&,24,Length[diff=Flatten[seqPosition[{#,Take[#,-2]}]]&[Map[a,Range[0,#]]]]<=1&];(#[[2]]-#[[1]])&[diff],{z,1,50}] (* _Peter J. C. Moses_, Apr 10 2013 *)
%K A224523 nonn
%O A224523 1,2
%A A224523 _Vladimir Shevelev_, Apr 09 2013
%E A224523 For n >= 9 the terms were calculated by _Peter J. C. Moses_, Apr 09 2013