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 A061641 Pure numbers in the Collatz (3x+1) iteration. Also called pure hailstone numbers. 18
 0, 1, 3, 6, 7, 9, 12, 15, 18, 19, 21, 24, 25, 27, 30, 33, 36, 37, 39, 42, 43, 45, 48, 51, 54, 55, 57, 60, 63, 66, 69, 72, 73, 75, 78, 79, 81, 84, 87, 90, 93, 96, 97, 99, 102, 105, 108, 109, 111, 114, 115, 117, 120, 123, 126, 127, 129, 132, 133, 135, 138, 141, 144, 145 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,3 COMMENTS Let {f(k,N), k=0,1,2,...} denote the (3x+1)-sequence with starting value N; a(n) denotes the smallest positive integer which is not contained in the union of f(k,0),...,f(k,a(n-1)). In other words a(n) is the starting value of the next '3x+1'-sequences in the sense that a(n) is not a value in any sequence f(k,N) with N < a(n). f(0,N)=N, f(k+1,N)=f(k,N)/2 if f(k,N) is even and f(k+1,N)=3*f(k,N)+1 if f(k,N) is odd. For all n, a(n) mod 6 is 0, 1 or 3. I conjecture that a(n)/n -> C=constant for n->oo, where C=2.311... The Collatz conjecture says that for all positive n, there exists k such that C_k(n) = 1. Shaw states [p. 195] that "A positive integer n is pure if its entire tree of preimages under the Collatz function C are greater than or equal to it; otherwise n is impure. Equivalently, a positive integer n is impure if there exists r nn || ! reached[[m]], If[m <= nn, reached[[m]] = True]; If[EvenQ[m], m = m/2, m = 3 m + 1]]]; nn = 200; reached = Table[False, {nn}]; t = {0, 1}; While[DoCollatz[t[[-1]]]; pos = Position[reached, False, 1, 1]; pos != {}, AppendTo[t, pos[[1, 1]]]]; t (* T. D. Noe, Jan 22 2013 *) PROG (PARI) firstMiss(A) = { my(i); if(#A == 0 || A > 0, return(0)); for(i = 1, A[#A] + 1, if(!setsearch(A, i), return(i))); }; iter(A) = { my(a = firstMiss(A)); while(!setsearch(A, a), A = setunion(A, Set([a])); a = if(a % 2, 3*a+1, a/2)); A; }; makeVec(m) = { my(v = [], A = Set([]), i); for(i = 1, m, v = concat(v, firstMiss(A)); if (i < m, A = iter(A))); v; }; makeVec(64) \\ Markus Sigg, Aug 08 2020 CROSSREFS Cf. A070165 (Collatz trajectories), A127633, A336938, A336938. See A177729 for a variant. Sequence in context: A026232 A286802 A297249 * A325443 A085359 A231664 Adjacent sequences:  A061638 A061639 A061640 * A061642 A061643 A061644 KEYWORD nice,nonn AUTHOR Frederick Magata (frederick.magata(AT)uni-muenster.de), Jun 14 2001 EXTENSIONS Edited by T. D. Noe and N. J. A. Sloane, Oct 16 2007 STATUS approved

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Last modified September 19 01:17 EDT 2020. Contains 337175 sequences. (Running on oeis4.)