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A193230 Start with 1; if even, divide by 2; if odd, add the next three primes.

%I A193230 #21 Oct 31 2017 20:34:46
%S A193230 1,11,60,30,15,74,37,168,84,42,21,104,52,26,13,72,36,18,9,50,25,122,
%T A193230 61,272,136,68,34,17,88,44,22,11,60,30,15,74,37,168,84,42,21,104,52,
%U A193230 26,13,72,36,18,9,50,25,122,61,272,136,68,34,17,88,44,22,11,60,30,15,74,37,168,84,42,21,104,52,26,13,72,36,18,9,50,25,122,61,272,136,68,34,17,88,44,22,11,60,30,15,74,37,168,84,42,21,104
%N A193230 Start with 1; if even, divide by 2; if odd, add the next three primes.
%C A193230 Trajectory of 1 under the map x -> A174221(x).
%C A193230 Periodic with period of length 30, starting at a(2) = 11.
%C A193230 Angelini conjectures that the orbit under A174221 becomes periodic for any initial value. He calls this the PrimeLatz conjecture, as tribute to L. Collatz, known for the 3n+1 conjecture.
%C A193230 It has been checked that the loop (11, ..., 22) (or (9, ..., 18), to start with the smallest element) 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. - _M. F. Hasler_, Oct 25 2017
%C A193230 It might have been more natural to start this sequence with offset 0. Since a(n) = a(n+30) from n = 2 on, this sequence consists essentially (except for the initial term) of the apparently unique "loop" of the "PrimeLatz" map A174221. It is used as such in related sequences A293978, ... - _M. F. Hasler_, Oct 31 2017
%H A193230 Eric Angelini, <a href="http://www.cetteadressecomportecinquantesignes.com/PrimeLatz.htm">The PrimeLatz conjecture</a>
%H A193230 E. Angelini, <a href="/A174221/a174221.pdf">The PrimeLatz Conjecture</a> [Cached copy, with permission]
%H A193230 <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 A193230 1 is odd;  we add to 1 the next 3 primes (2,3,5) and get 11
%e A193230 11 is odd;  we get 11+(13+17+19)=60
%e A193230 60 is even; we get 30
%e A193230 30 is even; we get 15
%e A193230 15 is odd;  we get 15+(17+19+23)=74
%e A193230 74 is even; we get 37
%e A193230 37 is odd;  we get 37+(41+43+47)=168
%e A193230 168 is even; we get 84
%e A193230 84 is even; we get 42
%e A193230 42 is even; we get 21
%e A193230 21 is odd;  we get 21+(23+29+31)=104
%e A193230 104 is even; we get 52
%e A193230 52 is even; we get 26
%e A193230 26 is even; we get 13
%e A193230 13 is odd;  we get 13+(17+19+23)=72
%e A193230 72 is even; we get 36
%e A193230 36 is even; we get 18
%e A193230 18 is even; we get 9
%e A193230 9 is odd;  we get 9+(11+13+17)=50
%e A193230 50 is even; we get 25
%e A193230 25 is odd;  we get 25+(29+31+37)=122
%e A193230 122 is even; we get 61
%e A193230 61 is odd;  we get 61+(67+71+73)=272
%e A193230 272 is even; we get 136
%e A193230 136 is even; we get 68
%e A193230 68 is even; we get 34
%e A193230 34 is even; we get 17
%e A193230 17 is odd;  we get 17+(19+23+29)=88
%e A193230 88 is even; we get 44
%e A193230 44 is even; we get 22
%e A193230 22 is even; we get 11... thus entering in a loop.
%e A193230 ...
%e A193230 (from Angelini's web page)
%t A193230 NestList[If[EvenQ@ #, #/2, Total@ Prepend[NextPrime[#, {1, 2, 3}], #]] &, 1, 101] (* _Michael De Vlieger_, Oct 25 2017 *)
%o A193230 (PARI) vector(100,i,t=if(i>1,A174221(t),1)) \\ _M. F. Hasler_, Oct 25 2017
%Y A193230 Cf. A174221, A293980, A293979 (orbit of 83), A293978 (orbit of 443), A293981 (orbit of 209).
%K A193230 nonn
%O A193230 1,2
%A A193230 _N. J. A. Sloane_, Jul 18 2011