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A000452 The greedy sequence of integers which avoids 3-term geometric progressions. 6
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

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.

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

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. - Nathan McNew, Jun 21 2014

LINKS

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000

B. Brown and D. Gordon, On sequences without geometric progressions, Math. Comp. 65 (1996), no. 216, 1749-1754.

R. Rankin, Sets of Integers Containing not more than a Given Number of Terms in Arithmetical Progression, Proc. Roy. Soc. Edinburgh Sect. A65 (1961), 332-344.

J. Riddell, Sets of integers containing no n terms in geometric progression, Glasgow Math. J. 10 (1969), 137-146.

F. Smarandache, Sequences of Numbers Involved in Unsolved Problems.

EXAMPLE

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.

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 *)

PROG

(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

CROSSREFS

Cf. A005836, A005117.

Sequence in context: A048964 A028749 A028764 * A047587 A285420 A028737

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

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Last modified July 21 04:20 EDT 2017. Contains 289632 sequences.