A084848 a(n) is the number of quadratic residues of A085635(n).

1, 2, 2, 3, 4, 4, 7, 8, 12, 14, 16, 16, 24, 28, 32, 42, 48, 48, 48, 64, 84, 96, 112, 144, 144, 176, 192, 192, 288, 336, 336, 504, 576, 576, 704, 864, 1008, 1056, 1152, 1232, 1152, 1344, 1728, 1920, 2016, 2016, 2352

Offset

1,2

Comments

Note that the terms are not all distinct.

Links

Keith F. Lynch, Table of n, a(n) for n = 1..200 (first 111 terms from Hugo Pfoertner).

Andreas Enge, William Hart, Fredrik Johansson, Short addition sequences for theta functions, arXiv:1608.06810 [math.NT], 2016-2018.

Formula

a(n) = A000224(A085635(n)). - Hugo Pfoertner, Aug 24 2018

Example

a(2)=2 because there are 2 different quadratic residues modulo 3, so 3 has 66.67% of quadratic residues density, while 2 has a 100%, so 3 has the least quadratic residues density up to 3.

Mathematica

Block[{s = Range[0, 2^15 + 1]^2, t}, t = Array[{#1/#2, #2} & @@ {#, Length@ Union@ Mod[Take[s, # + 1], #]} &, Length@ s - 1]; Map[t[[All, -1]][[FirstPosition[t[[All, 1]], #][[1]] ]] &, Union@ FoldList[Max, t[[All, 1]] ] ] ] (* Michael De Vlieger, Sep 10 2018 *)

Prog

(PARI) a000224(n)=my(f=factor(n)); prod(i=1, #f[, 1], if(f[i, 1]==2, 2^f[1, 2]\6+2, f[i, 1]^(f[i, 2]+1)\(2*f[i, 1]+2)+1)) \\ from Charles R Greathouse IV
r=2; for(k=1, 1e6, v=a000224(k); t=v/k; if(t<r, r=t; print1(v, ", "))) \\ Hugo Pfoertner, Aug 24 2018

Crossrefs

Cf. A000224, A085635.

Sequence in context: A198317 A036822 A056099 * A342330 A358911 A153937

Adjacent sequences: A084845 A084846 A084847 * A084849 A084850 A084851

Keyword

nonn

Author

Jose R. Brox (tautocrona(AT)terra.es), Jul 12 2003

Extensions

More terms from Jud McCranie, Jul 18 2003

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

Status

approved