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A293978 Start with 443; if even, divide by 2; if odd, add next three primes: Orbit of 443 under iterations of A174221, the "PrimeLatz" map.

%I A293978 #20 May 09 2021 11:20:37
%S A293978 443,1810,905,3642,1821,7322,3661,14682,7341,29410,14705,58858,29429,
%T A293978 117762,58881,235568,117784,58892,29446,14723,58932,29466,14733,58958,
%U A293978 29479,117990,58995,236012,118006,59003,236044,118022,59011,236084,118042,59021,236124,118062,59031,236198,118099,472536,236268
%N A293978 Start with 443; if even, divide by 2; if odd, add next three primes: Orbit of 443 under iterations of A174221, the "PrimeLatz" map.
%C A293978 Periodic with period 30, starting at a(9066) = 26 = A193230(14), see there for the next 30 elements which form the repeating part, a.k.a. loop.
%C A293978 Angelini conjectures that the orbit under A174221 becomes periodic for any initial value. He calls this the PrimeLatz conjecture (as a tribute to L. Collatz, known for the 3n+1 conjecture).
%C A293978 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, and the orbit of every positive integer <= 10^4 does end in this loop. See A293980 for the number of iterations required to reach an element of this loop.
%C A293978 Most small numbers (say, n < 1000) have very small orbits, and they converge into the above mentioned loop within a few iterations. The most remarkable exception is n = 83, whose orbit of 16210 elements is given in A293979. The second largest orbit (for "small" initial values) is that of 443, given here. It merges only near the end into that of 83, cf. Example section. Of course, the trajectory of any N = a(n)*2^k, e.g. 2*443 = 886, merges into the same orbit after k steps.
%H A293978 M. F. Hasler, <a href="/A293978/b293978.txt">Table of n, a(n) for n = 0..9096</a>
%H A293978 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/PrimeLatz.htm">The PrimeLatz conjecture</a>
%H A293978 <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 A293978 The initial value a(0) = 443 is odd, so we add to 443 the next 3 primes (449, 457 and 461) to get a(1) = 1810.
%e A293978 1810 is even, so we divide by 2 to get a(2) = 905, and so on.
%e A293978 After 2324 iterations, we get a(2324) = 4691214813495590981789155675545600. This is the largest value we will reach.
%e A293978 Since a(2324) is even, we divide by 2 to get a(2325), which is again even. This happens 12 times in a row; only after dividing by 2 for 13 times do we again reach an odd value, a(2337).
%e A293978 After 8853 iterations, we reach a(8853) = 3702 = A293979(15967). From here on, the tail of the orbit is the same as that of 83: 212 iterations later we get a(9065) = 3. Since this is odd, we add the next three primes (5, 7 and 11) to reach a(9066) = 26 = A193230(14). This is an element of the loop: 30 iterations later, we again get 26, and the sequence has become periodic.
%t A293978 NestList[If[EvenQ@ #, #/2, Total@ Prepend[NextPrime[#, {1, 2, 3}], #]] &, 83, 101]
%o A293978 (PARI) vector(100,i,t=if(i>1,A174221(t),443))
%Y A293978 Cf. A174221, A293980, A293979 (orbit of 83), A193230 (orbit of 1, includes the "loop" from the 2nd term of that sequence on).
%K A293978 nonn,look
%O A293978 0,1
%A A293978 _M. F. Hasler_, Oct 26 2017