A110924 - OEIS

%I #19 Aug 27 2020 21:28:08

%S 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,

%T 47,49,53,55,59,61,67,51,57,65,63,69,71,73,77,79,83,85,89,91,97,75,81,

%U 95,87,93,101,99,103,105,107,109,113,115,119,121,127,111,117,125,123,129

%N 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.

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

%H Robert Israel, <a href="/A110924/b110924.txt">Table of n, a(n) for n = 1..10000</a>

%e 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.

%p N:= 1000: # to get the first N terms

%p LCp:= proc(c,R,q)

%p local T,TM,m;

%p T:= select(t -> igcd(t,q) = 1, {$1 .. q-1});

%p m:= floor(c/q);

%p T:= map(`+`,T,m*q);

%p TM:= T minus R minus {$m*q .. c};

%p while TM = {} do

%p   T:= map(`+`,T,q);

%p   TM := T minus R;

%p od:

%p min(TM);

%p end proc;

%p A110924[1]:= 1: A110924[2]:= 2: A110924[3]:= 4: A110924[4]:= 5:

%p c:= 1: R:= {2,4,5}:

%p for n from 5 to N do

%p   A110924[n]:= LCp(c, R, A110924[n-1] + A110924[n-2]);

%p   if A110924[n] = c+2 then

%p     c:= c+2;

%p     while member(c+2,R) do c:= c+2 od:

%p     R:= select(`>`,R,c);

%p   else

%p     R:= R union {A110924[n]}

%p   fi;

%p od:

%p seq(A110924[n],n=1..N); # _Robert Israel_, May 09 2014

%t a[1] = 1; a[2] = 2; a[3] = 4;

%t 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]]]]];

%t Array[a, 100] (* _Jean-François Alcover_, Aug 26 2020 *)

%o (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

%K nonn

%O 1,2

%A _Leroy Quet_, Sep 23 2005

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