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%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
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