A085635 - OEIS

A085635

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.

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

(list; graph; refs; listen; history; text; internal format)

OFFSET

1,2

COMMENTS

After 2880, 3360 has exactly the same density (5%).

LINKS

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.

EXAMPLE

a(3)=4 because for B=4 the different quadratic residues are {0,1}, so S=2, the density is D_4=50%, which is smaller than D_2=100% and D_3=66.67%.

MATHEMATICA

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 *)

PROG

(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

CROSSREFS

Cf. A000224, A084848.

Sequence in context: A188217 A138926 A357501 * A077434 A076136 A064188

Adjacent sequences: A085632 A085633 A085634 * A085636 A085637 A085638

KEYWORD

nonn

AUTHOR