A110924 a(1) = 1, a(2) = 2; a(n) is smallest positive integer not among earlier terms of the sequence such that gcd(a(n), a(n-1) + a(n-2)) = 1.

1, 2, 4, 5, 7, 11, 13, 17, 19, 23, 25, 29, 31, 37, 3, 9, 35, 15, 21, 41, 27, 33, 43, 39, 45, 47, 49, 53, 55, 59, 61, 67, 51, 57, 65, 63, 69, 71, 73, 77, 79, 83, 85, 89, 91, 97, 75, 81, 95, 87, 93, 101, 99, 103, 105, 107, 109, 113, 115, 119, 121, 127, 111, 117, 125, 123, 129

Offset

1,2

Comments

2 and 4 are the only even terms in the sequence. Is every odd positive integer in the sequence?

Links

Robert Israel, Table of n, a(n) for n = 1..10000

Example

Of the positive integers not among the first 4 terms of the sequence, 7 is the smallest which is coprime to a(3) + a(4) = 4 + 5 = 9.

Maple

N:= 1000: # to get the first N terms
LCp:= proc(c, R, q)
local T, TM, m;
T:= select(t -> igcd(t, q) = 1, {$1 .. q-1});
m:= floor(c/q);
T:= map(`+`, T, m*q);
TM:= T minus R minus {$m*q .. c};
while TM = {} do
  T:= map(`+`, T, q);
  TM := T minus R;
od:
min(TM);
end proc;
A110924[1]:= 1: A110924[2]:= 2: A110924[3]:= 4: A110924[4]:= 5:
c:= 1: R:= {2, 4, 5}:
for n from 5 to N do
  A110924[n]:= LCp(c, R, A110924[n-1] + A110924[n-2]);
  if A110924[n] = c+2 then
    c:= c+2;
    while member(c+2, R) do c:= c+2 od:
    R:= select(`>`, R, c);
  else
    R:= R union {A110924[n]}
  fi;
od:
seq(A110924[n], n=1..N); # Robert Israel, May 09 2014

Mathematica

a[1] = 1; a[2] = 2; a[3] = 4;
a[n_] := a[n] = Module[{aa = Array[a, n-1], b = a[n-1] + a[n-2]}, For[k = 3, True, k += 2, If[FreeQ[aa, k], If[CoprimeQ[k, b], Return[k]]]]];
Array[a, 100] (* Jean-François Alcover, Aug 26 2020 *)

Prog

(PARI) { u=[2, 1]; c=3; s=u[1]+u[2]; m=Set(); m=setunion(m, [1]); m=setunion(m, [2]); print1(1, ", ", 2); for(k=1, 100, i=2; while(gcd(i, s)>1 || setsearch(m, i)!=0, i++); u[(c%2)+1] = i; c++; s=u[1]+u[2]; m=setunion(m, [i]); print1(i, ", ")) } \\ Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 25 2005

Crossrefs

Sequence in context: A147991 A033160 A350147 * A335402 A192590 A028289

Adjacent sequences: A110921 A110922 A110923 * A110925 A110926 A110927

Keyword

nonn

Author

Leroy Quet, Sep 23 2005

Extensions

More terms from Lambert Klasen (lambert.klasen(AT)gmx.net), Oct 25 2005

Status

approved