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A293979 Start with 83; if even, divide by 2; if odd, add next three primes: Orbit of 83 under iterations of A174221, the "PrimeLatz" map.

%I A293979 #16 Nov 11 2019 00:47:47
%S A293979 83,370,185,766,383,1570,785,3178,1589,6394,3197,12826,6413,25710,
%T A293979 12855,51536,25768,12884,6442,3221,12954,6477,25970,12985,51996,25998,
%U A293979 12999,52010,26005,104072,52036,26018,13009,52122,26061,104350,52175,208716,104358
%N A293979 Start with 83; if even, divide by 2; if odd, add next three primes: Orbit of 83 under iterations of A174221, the "PrimeLatz" map.
%C A293979 Periodic with period of length 30, starting at a(16180) = 26.
%C A293979 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 A293979 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 trajectory of every positive integer <= 10^4 does end in this loop.
%C A293979 See A293980 for the number of iterations required to reach an element of this loop, depending on the starting value.
%C A293979 Most small initial values have a very small orbit of few more than the 30 elements of the loop. N = 83 = a(0) is the most remarkable exception (having an orbit of 16180 + 30 elements), which motivates this sequence. Of course, the trajectory of any N = a(n)*2^k, e.g., 2*83 = 166, 4*83 = 332, 8*83 = 664, 2*370 = 740, ..., merges into the same orbit after k steps.
%C A293979 N = 443 = A293978(0) is another exception, with an orbit of 9066+30 elements (see A293978), and N = 209 also has a comparatively large orbit of 941 + 30 elements, distinct from those of 83 and 443.
%H A293979 M. F. Hasler, <a href="/A293979/b293979.txt">Table of n, a(n) for n = 0..16210</a>
%H A293979 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/PrimeLatz.htm">The PrimeLatz conjecture</a>
%H A293979 <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 A293979 The initial value a(0) = 83 is odd, so we add to 83 the next 3 primes (89, 97 and 101) to get a(1) = 370.
%e A293979 370 is even, so we divide by 2 to get a(2) = 185, and so on.
%e A293979 After 8337 iterations, we get a(8337) = 10780054699424618132644155893087038044817868609971935265882538442720. This is the largest value we will reach. Since this is even we divide by 2 to get a(8338).
%e A293979 The result a(8338) is again even, so we divide by 2 once more to get a(8339), and so on...
%e A293979 After iteration 16171, we reach a(16171) = 768. The next 8 iterations consist of dividing by 2, until we get a(16179) = 3. Since this is odd, we add the next three primes (5, 7 and 11) to reach a(16180) = 26 = A193230(14). This is an element of the loop: 30 iterations later, we get again 26, and the sequence has become periodic.
%t A293979 NestList[If[EvenQ@ #, #/2, Total@ Prepend[NextPrime[#, {1, 2, 3}], #]] &, 83, 101]
%o A293979 (PARI) vector(100,i,t=if(i>1,A174221(t),83))
%Y A293979 Cf. A174221, A293980, A293978 (orbit of 443), A193230 (orbit of 1).
%K A293979 nonn,look
%O A293979 0,1
%A A293979 _M. F. Hasler_, Oct 26 2017