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

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

1,2

Comments

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

Links

Alois P. Heinz, Table of n, a(n) for n = 1..10000

Math StackExchange, A sequence that avoids both arithmetic and geometric progression (2014)

Formula

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

Example

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

Prog

(PARI) {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
}
(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

Crossrefs

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

Sequence in context: A346759 A198331 A057518 * A153485 A244048 A309043

Adjacent sequences: A289203 A289204 A289205 * A289207 A289208 A289209

Keyword

nonn

Author

Roderick MacPhee, Jun 28 2017

Extensions

More terms from Alois P. Heinz, Jun 28 2017

Status

approved