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A293981 Start with 209; if even, divide by 2; if odd, add the next three primes: Trajectory of 209 under iterations of A174221, the "PrimeLatz" map.

%I A293981 #12 Nov 11 2019 00:49:41
%S A293981 209,870,435,1766,883,3588,1794,897,3634,1817,7318,3659,14680,7340,
%T A293981 3670,1835,7410,3705,14860,7430,3715,14894,7447,29814,14907,59698,
%U A293981 29849,119430,59715,238910,119455,477960,238980,119490,59745,239016,119508,59754,29877,119554,59777
%N A293981 Start with 209; if even, divide by 2; if odd, add the next three primes: Trajectory of 209 under iterations of A174221, the "PrimeLatz" map.
%C A293981 Periodic with period of length 30, starting at a(941) = 60.
%C A293981 Angelini conjectures that the trajectory under A174221 becomes periodic for any initial value. He called this the PrimeLatz conjecture (as tribute to L. Collatz, known for the 3n+1 conjecture).
%C A293981 It has been checked that the loop (9, ..., 18) (= A193230(19..48)) is the only loop (except for the fixed point 0) at least up to values not exceeding 10^8 (result due to _Hans Havermann_), and the trajectory of every positive integer <= 10^4 does end in this loop.
%C A293981 See A293980 for the number of iterations required to reach an element of this loop, depending on the starting value.
%C A293981 Most small initial values have a very small orbit of few more than the 30 elements of the loop. N = 83 = A293979(0) is the most remarkable exception, having an orbit of 16180 + 30 elements, cf. A293979.
%C A293981 N = 443 = A293978(0) is another exception, with an orbit of 9066+30 elements (see A293978).
%C A293981 N = 209 has the third largest genuinely different orbit among small initial values (of course, any N = 2^k*a(n) merges into the sequence a(n) after k steps), of 941 + 30 elements. This motivates the present sequence.
%C A293981 The fact that the loop is entered at a(941) = 60 = A193230(2), while the trajectories of 83 and 443 enter the loop at the term 26 = A193230(14), prove that this orbit is genuinely different from that of 83 and 443.
%C A293981 The horizontal rays in the graph correspond to factors of 2: division by 2 is one possible step, and for large numbers adding the next 3 primes roughly amounts to multiplying the value by 4, the prime gaps being "negligible".
%H A293981 M. F. Hasler, <a href="/A293981/b293981.txt">Table of n, a(n) for n = 0..971</a>
%H A293981 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/PrimeLatz.htm">The PrimeLatz conjecture</a>
%H A293981 <a href="/index/Rec#order_30">Index entries for linear recurrences with constant coefficients</a>, signature (0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1).
%e A293981 The initial value a(0) = 209 is odd, so we add to 209 the next 3 primes (211, 223 and 227) to get a(1) = 870.
%e A293981 a(1) = 870 is even, so we divide by 2 to get a(2) = 435, and so on.
%e A293981 After 667 iterations, we get a(667) = 517468668525760. This is the largest value we will reach. Since this is even we divide by 2 to get a(668).
%e A293981 The result a(668) is again even, so we divide by 2 once more to get a(669), and so on...
%e A293981 After iteration 935, we reach a(935) = 3840. The next 6 iterations consist of dividing by 2, until we get a(941) = 60 = A193230(2). This is an element of the loop: after dividing two more times by 2 and 28 other iterations later, we get again 60, and the sequence has become periodic.
%t A293981 NestList[If[EvenQ@ #, #/2, Total@ Prepend[NextPrime[#, {1, 2, 3}], #]] &, 83, 101]
%o A293981 (PARI) vector(100,i,t=if(i>1,A174221(t),209))
%Y A293981 Cf. A174221, A293980, A293979 (orbit of 83), A293978 (orbit of 443), A193230 (orbit of 1, essentially the apparently unique "loop" of A174221).
%K A293981 nonn,look
%O A293981 0,1
%A A293981 _M. F. Hasler_, Oct 31 2017