A000452 The greedy sequence of integers which avoids 3-term geometric progressions.

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

Offset: 1

Author

R. Muller

Keywords

• nonn
• nice

Comments

a(n) is the least integer greater than a(n-1) which does not form a 3-term 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. Adams-Watters, 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 3-term 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

Examples

			5 is included in the sequence because it is not part of any 3-term 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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• Megumi Asada, Bruce Fang, Eva Fourakis, Sarah Manski, Nathan McNew, Steven J. Miller, Gwyneth Moreland, Ajmain Yamin, and Sindy Xin Zhang, Avoiding 3-Term Geometric Progressions in Hurwitz Quaternions, Williams College (2023).
• Brienne E. Brown and Daniel M. Gordon, On sequences without geometric progressions, Math. Comp., Vol. 65, No. 216 (1996), pp. 1749-1754.
• 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. 332-344.
• J. Riddell, Sets of integers containing no n terms in geometric progression, Glasgow Math. J., Vol. 10, No. 2 (1969), pp. 137-146.
• Florentin Smarandache, Sequences of Numbers Involved in Unsolved Problems.

Crossrefs

Cf. A005836, A005117.

Programs

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}] (* Jean-François Alcover, Sep 21 2012 *)

PARI

find(v,x,final=#v,startAt=1)=my(m);while(final-startAt>1,m=(final+startAt)\2;if(v[m]>x,final=m-1,startAt=m));v[startAt]==x||v[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,i-1,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

Extensions

Corrected and extended by David W. Wilson, Aug 15 1996

Clarified and references added by Nathan McNew, Jun 21 2014