

A000452


The greedy sequence of integers which avoids 3term geometric progressions.


8



1, 2, 3, 5, 6, 7, 8, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 24, 26, 27, 29, 30, 31, 33, 34, 35, 37, 38, 39, 40, 41, 42, 43, 46, 47, 48, 51, 53, 54, 55, 56, 57, 58, 59, 61, 62, 65, 66, 67, 69, 70, 71, 73, 74, 77, 78, 79, 80, 81, 82, 83, 85, 86, 87, 88, 89, 91, 93, 94, 95, 97, 101
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OFFSET

1,2


COMMENTS

a(n) is the least integer greater than a(n1) which does not form a 3term geometric progression with 2 other terms already included in the sequence.
Equivalently, lexicographically earliest increasing sequence of nonnegative numbers that does contains the geometric mean of any pair of terms.  Keith F. Lynch, Jan 28 2018
A005117 is a subsequence.  Charles R Greathouse IV, Feb 24 2012
This is the sequence of all integers such that in the prime factorization, all exponents are in A005836.  Franklin T. AdamsWatters, Mar 04 2014, corrected by Charles R Greathouse IV, Sep 16 2015
From Nathan McNew, Jun 21 2014: (Start)
Rankin showed that the integers in this sequence have asymptotic density 0.71974..., where the exact value is given by an infinite product of zeta values.
Brown and Gordon showed that the sequence studied by Rankin is obtained by greedily including integers without creating a 3term geometric progression. (End)
The asymptotic density of this sequence is (1/zeta(2)) * Product_{k>=1} zeta(3^k)/zeta(2*3^k) = 0.7197452523101644272005259... (Rankin, 1961).  Amiram Eldar, Mar 20 2021


LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Brienne E. Brown and Daniel M. Gordon, On sequences without geometric progressions, Math. Comp., Vol. 65, No. 216 (1996), pp. 17491754.
Robert Alexander Rankin, Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression, Proc. Roy. Soc. Edinburgh Sect. A, Vol. 65, No. 4 (1961), pp. 332344.
J. Riddell, Sets of integers containing no n terms in geometric progression, Glasgow Math. J., Vol. 10, No. 2 (1969), pp. 137146.
Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems.


EXAMPLE

5 is included in the sequence because it is not part of any 3term geometric progression involving 1,2 or 3.
4 is not included because it forms a geometric progression with 1 and 2.
192 = 2^6 * 3. 6 is not in A005836 because it would make an arithmetic sequence with 0 and 3. So 2^0 * 3 = 3, 2^3 * 3 = 24, and 192 form a geometric progression, and thus 192 is not in the sequence.


MATHEMATICA

a[1] = 1; a[2] = 2; a[3] = 3; ok[n_, an_] := (aa = Append[ Select[ Array[a, n  1], Divisible[an, #] &], an]; m = Length[aa]; Do[ If[ aa[[j]]^2 == aa[[i]]*an, Return[False]], {i, 1, m  2}, {j, i + 1, m}]) =!= False; a[n_] := a[n] = For[an = a[n  1] + 1, True, an++, If[ ok[n, an], Return[an]]]; Table[ a[n], {n, 1, 75}] (* JeanFrançois Alcover, Sep 21 2012 *)


PROG

(PARI) find(v, x, final=#v, startAt=1)=my(m); while(finalstartAt>1, m=(final+startAt)\2; if(v[m]>x, final=m1, startAt=m)); v[startAt]==xv[final]==x
gp3(v, mx, n)=my(c=core(n, 1)); fordiv(c[2], d, if(d>1&&find(v, n/d, mx)&&find(v, n/d^2, mx), return(1))); 0
v=vector(100); v[1]=n=1; for(i=2, #v, while(gp3(v, i1, n++), ); v[i]=n); v \\ Charles R Greathouse IV, Feb 24 2012
(PARI) is(n)=#select(n>while(n, if(n%3>1, return(1)); n\=3); 0, factor(n)[, 2])==0 \\ Charles R Greathouse IV, Sep 16 2015


CROSSREFS

Cf. A005836, A005117.
Sequence in context: A337050 A304364 A336590 * A307295 A047587 A285420
Adjacent sequences: A000449 A000450 A000451 * A000453 A000454 A000455


KEYWORD

nonn,nice


AUTHOR

R. Muller


EXTENSIONS

Corrected and extended by David W. Wilson, Aug 15 1996
Clarified and references added by Nathan McNew, Jun 21 2014


STATUS

approved



