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A085635
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Compute S, the number of different quadratic residues modulo B for every base B. If the density S/B is smaller for B than for every B' less than B, then B is added to the sequence.
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6
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1, 3, 4, 8, 12, 16, 32, 48, 80, 96, 112, 144, 240, 288, 336, 480, 560, 576, 720, 1008, 1440, 1680, 2016, 2640, 2880, 3600, 4032, 5040, 7920, 9360, 10080, 15840, 18480, 20160, 25200, 31680, 37440, 39600, 44352, 50400, 55440, 65520, 85680, 95760
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OFFSET
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1,2
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COMMENTS
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After 2880, 3360 has exactly the same density (5%).
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LINKS
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Keith F. Lynch, Table of n, a(n) for n = 1..200 (terms 1..111) from Hugo Pfoertner)
Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.
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EXAMPLE
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a(3)=4 because for B=4 the different quadratic residues are {0,1}, so S=2, the density is D_4=50%, that is smaller than D_2=100% and D_3=66.67%.
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MATHEMATICA
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Block[{s = Range[0, 2^14 + 1]^2, t}, t = Array[#/Length@ Union@ Mod[Take[s, # + 1], #] &, Length@ s - 1]; Map[FirstPosition[t, #][[1]] &, Union@ FoldList[Max, t]]] (* Michael De Vlieger, Sep 10 2018 *)
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PROG
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(PARI) r=-1; for(n=1, 1e6, t=1-sum(k=1, n, issquare(Mod(k, n)))/n; if(t>r, r=t; print1(n", "))) \\ Charles R Greathouse IV, Jul 15 2011
(PARI) sq1(m)=sum(n=0, m-1, issquare(Mod(n, m)))
sq(n, f=factor(n))=prod(i=1, #f~, my(p=f[i, 1], e=f[i, 2]); if(e>1, sq1(p^e), p\2+1))
r=2; for(n=1, 1e6, t=sq(n)/n; if(t<r, r=t; print1(n", "))) \\ Charles R Greathouse IV, Mar 30 2018
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CROSSREFS
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Cf. A000224, A084848.
Sequence in context: A190158 A188217 A138926 * A077434 A076136 A064188
Adjacent sequences: A085632 A085633 A085634 * A085636 A085637 A085638
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KEYWORD
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nonn
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AUTHOR
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Jose R. Brox (tautocrona(AT)terra.es), Jul 10 2003
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EXTENSIONS
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More terms from Jud McCranie, Jul 12 2003
a(1) and PARI programs corrected by Hugo Pfoertner, Aug 23 2018
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STATUS
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approved
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