Year-end appeal: Please make a donation to the OEIS Foundation to support ongoing development and maintenance of the OEIS. We are now in our 61st year, we have over 378,000 sequences, and we’ve reached 11,000 citations (which often say “discovered thanks to the OEIS”).
%I #37 Oct 16 2015 09:58:00
%S 34,59,44,79,56,107,75,54,103,72,151,102,233,153,104,239,156,397,253,
%T 165,112,263,171,116,271,176,457,290,829,512,1619,974,3469,2044,8123,
%U 4696,20879,11861,6807,3952,17159,9786,47459
%N Rocket Sequence 34: a(0)=34, a(n) = A073846(a(n-1)).
%C A073846(n) is defined as follows: if n = 2m for some integer m, A073846(n) is the m-th prime, if n = 2m-1 for some integer m, A073846(n) is the m-th nonprime.
%C Consider the (totally) ordered set {n, A073846(n), A073846(A073846(n))...} and let us append to this the ordered set {...b(b(b(n))),b(b(n)),b(n)} where b(m) = A073898(m) is the inverse of A073846. Let us call the result R#(n). It is clear that if m is a value in R#(n), R#(m) is just R#(n) with a different offset. Therefore, unless there is a need to do otherwise, let us denote each sequence by its lowest value. {a(n)} when extended to all integers (the last few unlisted values are ... 36, 61, 45, 34) is R#(34).
%C A given sequence c(n) can be one of two kinds. It can either be periodic with c(m) = c(0) for some m, or it can include infinitely many distinct values. R#(n) is finite for all n<34. However, this sequence has been checked up to a(86) = 1091595086717 without reaching 34. Instead it seems to be slowly climbing in value in both the negative and positive directions. Hence, its period is either extremely large or nonexistent (infinite). I conjecture that the latter is the case. Thus I dubbed the sequence "Rocket" because, as opposed to the "Hailstone" sequences, it never seems to "fall".
%H Chayim Lowen, <a href="/A261314/b261314.txt">Table of n, a(n) for n = 0..86</a>
%F a(n+1) = A073846(a(n)) = A018252(ceiling(a(n)/2))*A000035(a(n)) + A000040(ceiling(a(n)/2))*A059841(a(n)).
%F a(n-1) = A073898(a(n)) = 2*A010051(a(n))*A000720(a(n)) + (1-A010051(a(n)))*(2*A018252(a(n))-1).
%e a(1) = A073846(a(0)) = A073846(34) = 59.
%t f[n_, lim_] := Block[{p = Prime@ Range@ PrimePi@ lim, c, s, a = {34}}, c = Complement[Range@ lim, p]; s = Riffle[Take[c, Length@ p], p]; Do[AppendTo[a, s[[a[[k]]]]], {k, n}]; a]; f[48, 10000000] (* _Michael De Vlieger_, Aug 26 2015, after _Harvey P. Dale_ at A073846 *)
%Y Cf. A073846.
%K nonn
%O 0,1
%A _Chayim Lowen_, Aug 14 2015