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.

83, 370, 185, 766, 383, 1570, 785, 3178, 1589, 6394, 3197, 12826, 6413, 25710, 12855, 51536, 25768, 12884, 6442, 3221, 12954, 6477, 25970, 12985, 51996, 25998, 12999, 52010, 26005, 104072, 52036, 26018, 13009, 52122, 26061, 104350, 52175, 208716, 104358

Offset: 0

Author

M. F. Hasler, Oct 26 2017

Keywords

• nonn
• look

Comments

Periodic with period of length 30, starting at a(16180) = 26.

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, 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 = 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.

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.

Examples

			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.
370 is even, so we divide by 2 to get a(2) = 185, and so on.
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).
The result a(8338) is again even, so we divide by 2 once more to get a(8339), and so on...
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.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..16210
• 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, A293978 (orbit of 443), A193230 (orbit of 1).

Programs

Mathematica

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

PARI

vector(100,i,t=if(i>1,A174221(t),83))