A261314 Rocket Sequence 34: a(0)=34, a(n) = A073846(a(n-1)).

34, 59, 44, 79, 56, 107, 75, 54, 103, 72, 151, 102, 233, 153, 104, 239, 156, 397, 253, 165, 112, 263, 171, 116, 271, 176, 457, 290, 829, 512, 1619, 974, 3469, 2044, 8123, 4696, 20879, 11861, 6807, 3952, 17159, 9786, 47459

Offset: 0

Author

Chayim Lowen, Aug 14 2015

Keywords

• nonn

Comments

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.

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).

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".

Examples

			a(1) = A073846(a(0)) = A073846(34) = 59.
		

Links

• Chayim Lowen, Table of n, a(n) for n = 0..86

Crossrefs

Cf. A073846.

Programs

Mathematica

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 *)

Formula

a(n+1) = A073846(a(n)) = A018252(ceiling(a(n)/2))*A000035(a(n)) + A000040(ceiling(a(n)/2))*A059841(a(n)).

a(n-1) = A073898(a(n)) = 2*A010051(a(n))*A000720(a(n)) + (1-A010051(a(n)))*(2*A018252(a(n))-1).