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%I #48 Jan 31 2024 07:58:12
%S 4,6,9,13,2,5,2,6,9,13,2,7,2,8,12,17,4,6,9,13,2,9,13,2,10,14,20,28,37,
%T 6,9,13,2,11,4,6,9,13,2,12,17,4,6,9,13,2,13,2,14,20,28,37,6,9,13,2,15,
%U 21,29,6,9,13,2,16,22,30,40,52,67,6,9,13,2,17,4,6,9,13,2
%N Irregular triangle read by rows where row n is the trajectory starting from n and ending with 2 of the map x -> A368241(x).
%C It is conjectured that every starting n reaches 2 eventually. (If not then the sequence has an infinite final row.)
%C Map A368241(x) decreases to the prime gap x-prevprime(x) when x is prime, or increases to x+primepi(x) otherwise, and will reach 2 when x is the greater of a twin prime pair (A006512, preceding prime gap 2).
%C Prime gaps and x+primepi(x) may become large, but if the twin prime conjecture is true then there would be large twin primes they might reach too.
%F T(n,0) = n.
%F T(n,k) = A368241(T(n,k-1)) for k >= 1.
%e Table T(n,k) begins:
%e n\k 0 1 2 3 4 5 6 7 8 9
%e --------------------------------------------
%e 4: 4 6 9 13 2
%e 5: 5 2
%e 6: 6 9 13 2
%e 7: 7 2
%e 8: 8 12 17 4 6 9 13 2
%e 9: 9 13 2
%e 10: 10 14 20 28 37 6 9 13 2
%e 11: 11 4 6 9 13 2
%e 12: 12 17 4 6 9 13 2
%e 13: 13 2
%e 14: 14 20 28 37 6 9 13 2
%e 15: 15 21 29 6 9 13 2
%e 16: 16 22 30 40 52 67 6 9 13 2
%e 17: 17 4 6 9 13 2
%e 18: 18 25 34 45 59 6 9 13 2
%e 19: 19 2
%e 20: 20 28 37 6 9 13 2
%o (PARI) row(n) = my(list=List(n)); while(n!=2, n = if (isprime(n), n - precprime(n-1), n + primepi(n)); listput(list, n)); Vec(list); \\ _Michel Marcus_, Dec 17 2023
%Y Cf. A368241.
%Y Cf. A000720, A005171, A010051, A006512.
%K nonn,tabf
%O 4,1
%A _Hendrik Kuipers_, Dec 16 2023