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

1,2

Comments

It appears that starting at 64, the sequence follows the pattern, 2^(2n) + {0, 16, 21, 36, 64, 66, 84, 96, 144, 168} * 2^(2n-6) for n >= 3. - T. D. Noe, Jun 20 2012

Intersection of A001481 and A004214. - Ralf Stephan, Mar 28 2014

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

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

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 *)

Prog

(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
(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

Crossrefs

Sequence in context: A239517 A231056 A325363 * A191985 A126026 A199425

Adjacent sequences: A000546 A000547 A000548 * A000550 A000551 A000552

Keyword

nonn

Author

N. J. A. Sloane and J. H. Conway

Extensions

Extended by T. D. Noe, Jun 20 2012

Status

approved