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.

443, 1810, 905, 3642, 1821, 7322, 3661, 14682, 7341, 29410, 14705, 58858, 29429, 117762, 58881, 235568, 117784, 58892, 29446, 14723, 58932, 29466, 14733, 58958, 29479, 117990, 58995, 236012, 118006, 59003, 236044, 118022, 59011, 236084, 118042, 59021, 236124, 118062, 59031, 236198, 118099, 472536, 236268

Offset: 0

Author

M. F. Hasler, Oct 26 2017

Keywords

• nonn
• look

Comments

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.

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).

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.

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.

Examples

			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.
1810 is even, so we divide by 2 to get a(2) = 905, and so on.
After 2324 iterations, we get a(2324) = 4691214813495590981789155675545600. This is the largest value we will reach.
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).
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.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 0..9096
• 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), A193230 (orbit of 1, includes the "loop" from the 2nd term of that sequence on).

Programs

Mathematica

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

PARI

vector(100,i,t=if(i>1,A174221(t),443))