A000549 Numbers that are the sum of 2 squares but not sum of 3 nonzero squares.
1, 2, 4, 5, 8, 10, 13, 16, 20, 25, 32, 37, 40, 52, 58, 64, 80, 85, 100, 128, 130, 148, 160, 208, 232, 256, 320, 340, 400, 512, 520, 592, 640, 832, 928, 1024, 1280, 1360, 1600, 2048, 2080, 2368, 2560, 3328, 3712, 4096, 5120, 5440, 6400, 8192, 8320, 9472, 10240
Offset: 1
Keywords
Links
- Robert Israel, Table of n, a(n) for n = 1..113 (first 100 terms from T. D. Noe)
- Index entries for sequences related to sums of squares
Programs
-
MATLAB
Asq = zeros(1,10^7); Asq( [0:3162] .^ 2 + 1) = 1; Psq = zeros(1,10^7); Psq( [1:3162] .^ 2 + 1) = 1; As2 = conv(Asq,Asq); As2 = min(1, As2(1:10^7)); Ps2 = conv(Psq,Psq); Ps2 = min(1, Ps2(1:10^7)); As3 = conv(Asq,As2); As3 = min(1, As3(1:10^7)); Ps3 = conv(Psq,Ps2); Ps3 = min(1, Ps3(1:10^7)); D = As3 - Ps3; A000549 = find(D(2:end)) % Robert Israel, Mar 26 2014
-
Maple
# this requires Maple 17 or later m:= 5000: # to find all entries <= m^2 N:= m^2: with(SignalProcessing): with(ArrayTools): A1:= Array(0..N,datatype=float[8]): for i from 1 to m do A1[i^2]:= 1 end do: B:= Convolution(A1,A1): A2:= Array(0..N,datatype=float[8]): Copy(N+1,B,A2): All1:= Array(0..N,datatype=float[8],fill=1): MinimumEvery(A2,All1,container=A2): B:= Convolution(A1,A2): A3:= Array(0..N,datatype=float[8]): Copy(N+1,B,A3); MinimumEvery(A3,All1,container=A3); MaximumEvery(A1,A2,container=A2); B:= evalhf(map(round,A3-A2)); MinimumEvery(B,Array(0..N,datatype=float[8]),container=B); R:= ArrayTools[SearchArray](B); A000549:= map(`-`,convert(R,list),1); # Robert Israel, Mar 28 2014
-
Mathematica
A000549 = Reap[For[n=1, n <= 10^5, n++, If[SquaresR[2, n] != 0, If[Union[ PowersRepresentations[n, 3, 2][[All, 1]]] == {0}, Print[n]; Sow[n]]]] ][[2, 1]] (* Jean-François Alcover, Feb 09 2016 *)
-
PARI
istwo(n)=if(n<3,return(n>=0)); my(f=factor(n>>valuation(n,2))); for(i=1,#f~, if(f[i,2]%2&&f[i,1]%4==3, return(0))); 1 is(n)=if(!istwo(n),return(0)); if(n<3, return(1)); for(x=sqrtint(n\3),sqrtint(n-2), my(t=n-x^2); for(y=sqrtint(t\2),sqrtint(t-1), if(t&&issquare(t-y^2), return(0)))); 1 \\ Charles R Greathouse IV, Jan 28 2016
Formula
Empirical g.f.: x*(1 +2*x +4*x^2 +5*x^3 +8*x^4 +10*x^5 +13*x^6 +16*x^7 +20*x^8 +25*x^9 +28*x^10 +29*x^11 +24*x^12 +32*x^13 +26*x^14 +24*x^15 +28*x^16 +21*x^17 +20*x^18 +28*x^19 +2*x^20) / ((1 -2*x^5)*(1 +2*x^5)). - Colin Barker, Feb 09 2016
Extensions
Extended by T. D. Noe, Jun 20 2012
Comments