A224853 -id:A224853 - cp's OEIS frontend

A225571 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic or geometric subsequence.

0, 1, 3, 4, 10, 11, 13, 14, 29, 30, 32, 33, 38, 39, 41, 42, 85, 86, 88, 89, 94, 95, 97, 98, 112, 113, 115, 116, 122, 123, 125, 238, 248, 251, 252, 255, 257, 260, 261, 273, 275, 278, 279, 287, 288, 292, 330, 331, 334, 335

Offset: 1

Giovanni Resta, Jul 29 2013

This sequence diverges from A224853 at 477th term. Here a(477) = 17380, while A224853(477) = 17408.

			After terms 0, 1, 3, 4 have been added, the terms 5,...,9 are forbidden by subsequences (3,4,5), (0,3,6), (1,4,7), (0,4,8) and (1,3,9) so the next term is 10.

Giovanni Resta, Table of n, a(n) for n = 1..10000

Cf. A224853, A003278, A005836, A000452.

seq = {0, 1}; bad[n_] := Catch[ Do[If[MemberQ[seq, (n + e)/2], Throw@True], {e, seq}];  Do[If[MemberQ[seq, Sqrt[n*e]], Throw@True], {e, Rest@ seq}]; False]; While[Length[seq] < 100, x = Last[seq]+1; While[bad[x], x++]; AppendTo[seq, x]]; seq

A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3.

Original entry on oeis.org

1, 2, 5, 6, 12, 13, 15, 16, 32, 33, 35, 39, 40, 42, 56, 81, 84, 85, 88, 90, 93, 94, 108, 109, 113, 115, 116, 159, 189, 207, 208, 222, 223, 232, 235, 240, 243, 244, 249, 250, 252, 259, 267, 271, 289, 304, 314, 318, 325, 340, 342, 397, 504, 508, 511, 531, 549

Offset: 1

Roderick MacPhee, Jun 28 2017

nonn

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.

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

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

Alois P. Heinz, Table of n, a(n) for n = 1..10000
Math StackExchange, A sequence that avoids both arithmetic and geometric progression (2014)

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

{my(a=[1,2]);
for(x=3,100,
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
}

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

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

More terms from Alois P. Heinz, Jun 28 2017

cp's OEIS Frontend

A225571 Lexicographically earliest sequence of nonnegative integers which does not contain a three-term arithmetic or geometric subsequence.

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