A289206 - OEIS

%I #39 Jul 03 2017 09:14:23

%S 1,2,5,6,12,13,15,16,32,33,35,39,40,42,56,81,84,85,88,90,93,94,108,

%T 109,113,115,116,159,189,207,208,222,223,232,235,240,243,244,249,250,

%U 252,259,267,271,289,304,314,318,325,340,342,397,504,508,511,531,549

%N Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

%C By avoiding arithmetic progressions, at most 2/3 of the numbers up to a(n) are in the sequence. The sequence doesn't contain 3 consecutive powers in arithmetic progression for any base c.

%C Where a(n)+1 = a(n+1): 1, 3, 5, 7, 9, 12, 17, 21, 23, 26, 30, 32, 37, 39, etc. - _Robert G. Wilson v_, Jul 02 2017

%H Alois P. Heinz, <a href="/A289206/b289206.txt">Table of n, a(n) for n = 1..10000</a>

%H Math StackExchange, <a href="https://math.stackexchange.com/questions/858918/a-sequence-that-avoids-both-arithmetic-and-geometric-progressions">A sequence that avoids both arithmetic and geometric progression</a> (2014)

%F a(n) >= 3n/2 for n > 2.

%e 5 is in the sequence because 1,2,5 is neither an arithmetic progression nor a geometric progression.

%o (PARI) {my(a=[1,2]);

%o for(x=3,100,

%o if(#select(r->#select(q->q==2*r,b)==0,b=vecsort(apply(r->x-r,a)))==#a && #select(r->#select(q->q==r^2,b)==0,b=vecsort(apply(r->x/r,a)))==#a,a=concat(a,x)));a

%o }

%o (PARI) first(n)=my(v=vector(n)); v[1]=1; for(k=2,n, my(avoid=List(),t,last=v[k-1]); for(i=2,k-1, for(j=1,i-1, t=2*v[i]-v[j]; if(t>last, listput(avoid, t)); if(denominator(t=v[i]^2/v[j])==1 && t>last, listput(avoid,t)))); avoid=Set(avoid); for(i=v[k-1]+1,v[k-1]+#avoid+1, if(!setsearch(avoid,i), v[k]=i; break))); v \\ _Charles R Greathouse IV_, Jun 29 2017

%Y Cf. A000452, A003278, A005836, A224853, A225571.

%K nonn

%O 1,2

%A _Roderick MacPhee_, Jun 28 2017

%E More terms from _Alois P. Heinz_, Jun 28 2017