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.

209, 870, 435, 1766, 883, 3588, 1794, 897, 3634, 1817, 7318, 3659, 14680, 7340, 3670, 1835, 7410, 3705, 14860, 7430, 3715, 14894, 7447, 29814, 14907, 59698, 29849, 119430, 59715, 238910, 119455, 477960, 238980, 119490, 59745, 239016, 119508, 59754, 29877, 119554, 59777

Offset: 0

Author

M. F. Hasler, Oct 31 2017

Keywords

• nonn
• look

Comments

Periodic with period of length 30, starting at a(941) = 60.

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).

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.

See A293980 for the number of iterations required to reach an element of this loop, depending on the starting value.

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.

N = 443 = A293978(0) is another exception, with an orbit of 9066+30 elements (see A293978).

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.

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.

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".

Examples

			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.
a(1) = 870 is even, so we divide by 2 to get a(2) = 435, and so on.
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).
The result a(668) is again even, so we divide by 2 once more to get a(669), and so on...
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.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..971
• Eric Angelini, The PrimeLatz conjecture
• Index entries for linear recurrences with constant coefficients, 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).

Crossrefs

Cf. A174221, A293980, A293979 (orbit of 83), A293978 (orbit of 443), A193230 (orbit of 1, essentially the apparently unique "loop" of A174221).

Programs

Mathematica

NestList[If[EvenQ@ #, #/2, Total@ Prepend[NextPrime[#, {1, 2, 3}], #]] &, 83, 101]

PARI

vector(100,i,t=if(i>1,A174221(t),209))