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A230490
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Size of largest subset of [1..n] containing no three terms in a geometric progression with integer ratio.
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1
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1, 2, 3, 3, 4, 5, 6, 7, 7, 8, 9, 10, 11, 12, 13, 14, 15, 15, 16, 16, 17, 18, 19, 20, 21, 22, 23, 23, 24, 25, 26, 26, 27, 28, 29, 30, 31, 32, 33, 33, 34, 35, 36, 36, 37, 38, 39, 39, 40, 41, 42, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 52, 52, 53, 54, 55, 55, 56, 57, 58, 59, 60, 61, 62, 62, 63, 64, 65, 65, 66, 67, 68, 69, 70, 71, 72, 73, 74, 75, 76, 76, 77, 78, 79, 79, 80, 81, 81, 81
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OFFSET
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1,2
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COMMENTS
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McNew proves that if n is sufficiently large, then the n-th term is between 0.818n and 0.820n. - Kevin O'Bryant, Aug 17 2015
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LINKS
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EXAMPLE
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The integers [1..9] include the three geometric progressions (1,2,4) (2,4,8) and (1,3,9), which cannot all be precluded with any 1 exclusion, but 2 exclusions suffice. Thus the size of the largest subsets of [1..9] free of integer ratio geometric progressions is 7.
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PROG
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(PARI) ok(v)=for(i=3, #v, my(k=v[i]); fordiv(core(k, 1)[2], d, if(d>1 && setsearch(v, k/d) && setsearch(v, k/d^2), return(0)))); 1
a(n)=my(v=select(k->4*k>n&&issquarefree(k), vector(n, i, i)), u=setminus(vector(n, i, i), v), r, H); for(i=1, 2^#u-1, H=hammingweight(i); if(H>r && ok(vecsort(concat(v, vecextract(u, i)), , 8)), r=H)); #v+r \\ Charles R Greathouse IV, Oct 20 2013
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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