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

Author

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

Keywords

• nonn

Comments

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

Examples

			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%.
		

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.

Crossrefs

Cf. A000224, A084848.

Programs

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 *)

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(tCharles R Greathouse IV, Mar 30 2018

Extensions

More terms from Jud McCranie, Jul 12 2003

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