A140490 Trajectory of 1 under repeated application of the map: n -> n + third-smallest number that does not divide n.

1, 5, 9, 14, 19, 23, 27, 32, 38, 43, 47, 51, 56, 62, 67, 71, 75, 81, 86, 91, 95, 99, 104, 110, 116, 122, 127, 131, 135, 141, 146, 151, 155, 159, 164, 170, 176, 182, 187, 191, 195, 201, 206, 211, 215, 219, 224, 230, 236, 242, 247, 251, 255, 261, 266, 271, 275, 279, 284, 290, 296

Offset: 1

Author

Jacques Tramu, Jun 25 2008

Keywords

• nonn

Comments

Suggested by Eric Angelini.

Links

• Robert Israel, Table of n, a(n) for n = 1..10000
• Index entries for linear recurrences with constant coefficients, signature (1,0,0,0,0,0,0,0,0,0,0,1,-1).

Crossrefs

Cf. A140485, A140486, A140487, A140488, A140489 (second-smallest sequences).

Cf. A140491, A140492, A140493, A140494 (third-smallest sequences).

Programs

Maple

f:= proc(n) local k,count;
  count:= 0;
  for k from 2 do
    if n mod k <> 0 then count:= count+1; if count = 3 then return n+k fi fi
  od
end proc:
R:= 1: x:= 1:
for i from 1 to 100 do x:= f(x); R:= R, x od:
R; # Robert Israel, Jan 17 2023

Mathematica

NestList[#+Complement[Range[#+50],Divisors[#]][[3]]&,1,60] (* Harvey P. Dale, Apr 21 2022 *)

PARI

third(n) = {my(nb = 0, k = 1); while (nb != 3, if (n % k, nb++); if (nb != 3, k++);); k;}
f(n) = n + third(n);
lista1(nn) = {a = 1; print1(a, ", "); for (n=2, nn, newa = f(a); print1(newa, ", "); a = f(a););} \\ Michel Marcus, Oct 04 2018

Formula

a(n+12) = a(n) + 60 for n >= 13. - Robert Israel, Jan 17 2023

From Chai Wah Wu, Nov 14 2024: (Start)

A140490-A140493 all converge to the same trajectory.

a(n) = a(n-1) + a(n-12) - a(n-13) for n > 25.

G.f.: x*(x^24 + 2*x^23 + x^22 - x^21 - 2*x^20 + x^18 + 2*x^17 - x^16 - x^15 + x^14 + 2*x^13 + 4*x^12 + 4*x^11 + 4*x^10 + 5*x^9 + 6*x^8 + 5*x^7 + 4*x^6 + 4*x^5 + 5*x^4 + 5*x^3 + 4*x^2 + 4*x + 1)/(x^13 - x^12 - x + 1). (End)

Extensions

More terms from Michel Marcus, Oct 04 2018