A140485 Trajectory of 1 under repeated application of the map: n -> n + second-smallest number that does not divide n.

1, 4, 9, 13, 16, 21, 25, 28, 33, 37, 40, 46, 50, 54, 59, 62, 66, 71, 74, 78, 83, 86, 90, 97, 100, 106, 110, 114, 119, 122, 126, 131, 134, 138, 143, 146, 150, 157, 160, 166, 170, 174, 179, 182, 186, 191, 194, 198, 203, 206, 210, 218, 222, 227, 230, 234, 239, 242, 246

Offset

1,2

Comments

a(n) mod 420 is periodic. This can be seen because if b = A396771(a(n)) does not divide a(n) and also not divide a(n)+420 then b will also not divide a(n)+j*420 for some j under the restriction that gcd(420, b) does not divide k. This last condition is always met as we observe that A396771(a(n) mod 420) <= 8 for all n and thus gcd(420, b) = b for b < 8. - Thomas Scheuerle, Jun 06 2026

Links

Michael S. Branicky, Table of n, a(n) for n = 1..10000

Index entries for linear recurrences with constant coefficients, signature (1, 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, 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, 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, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, -1).

Formula

a(n + 98) = a(n) + 420 for n >= 10. - David Radcliffe, Jun 04 2026

a(n) = a(n-1) + a(n-98) - a(n-99) for n >= 108. - Thomas Scheuerle, Jun 04 2026

Example

The numbers that do not divide 4 are 3, 5, 6, 7, ..., so a(3) = 4+5 = 9.
Here are the beginnings of the trajectories of some small numbers:
...1--4--9---13--16--21--25--28--32--37--40---
.............|...................|...|
......5--8---+...............29--+...|
.....................................|
...2--6--11--14--18--23--26--30------+
.............|...........|...........|
...3--7--10--+...........|.......33--+
.........................|
.............12--19--22--+
.................|.......|
.............15--+.......|
.........................|
.................17--20--+
..............................................
.........................24--31--34--38--42---
.............................|.......|
.........................27--+...35--+

Mathematica

f[n_] := (k = 1; s = {}; While[ True, k++; If[ !Divisible[n, k], AppendTo[s, k]]; If[Length[s] == 2, Break[]]]; n + Last[s]); NestList[f, 1, 58] (* Jean-François Alcover, Oct 05 2011 *)
(* Alternative: *)
NestList[#+Complement[Range[100], Divisors[#]][[2]]&, 1, 60] (* Harvey P. Dale, Apr 27 2012 *)

Prog

(Python)
from itertools import islice
def f(x):
    nd1 = next(k for k in range(2, x+2) if x%k)
    return x + next(k for k in range(nd1+1, x+3) if x%k)
def agen(x=1): # generator of terms
    while True: yield x; x = f(x)
print(list(islice(agen(), 59))) # Michael S. Branicky, Jun 03 2026

Crossrefs

Cf. A140486, A140487, A140488, A140489, A140595. ( based on second-smallest number )

Cf. A140491, A140492, A140493, A140494. ( based on third-smallest number )

Cf. A396771.

Sequence in context: A141000 A312878 A312879 * A312880 A312881 A312882

Adjacent sequences: A140482 A140483 A140484 * A140486 A140487 A140488

Keyword

nonn,easy

Author

Eric Angelini, Jun 25 2008

Extensions

More terms from Stefan Steinerberger, Jul 01 2008

Status

approved