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A289206 Greedy strictly increasing sequence starting at a(1)=1 avoiding both arithmetic and geometric progressions of length 3. 1
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 (list; graph; refs; listen; history; text; internal format)
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: A086334 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

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Last modified September 17 19:16 EDT 2019. Contains 327137 sequences. (Running on oeis4.)