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

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

Jose R. Brox (tautocrona(AT)terra.es), Jul 10 2003

Extensions

More terms from Jud McCranie, Jul 12 2003

a(1) and PARI programs corrected by Hugo Pfoertner, Aug 23 2018

Status

approved