A001481 -id:A001481 - cp's OEIS frontend

A303639 Number of ways to write n as a^2 + b^2 + binomial(2c+1,c) + binomial(2d+1,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

0, 1, 1, 2, 1, 3, 2, 2, 1, 2, 3, 3, 3, 3, 4, 2, 2, 2, 3, 4, 4, 5, 2, 4, 1, 2, 3, 3, 5, 3, 5, 1, 3, 1, 1, 6, 3, 8, 3, 6, 2, 4, 4, 2, 7, 5, 6, 2, 5, 2, 4, 5, 4, 8, 4, 7, 2, 4, 1, 3, 6, 4, 7, 3, 5, 2, 4, 2, 4, 9, 5, 6, 2, 6, 4, 5, 4, 7, 5, 2

Offset: 1

Zhi-Wei Sun, Apr 27 2018

nonn

Conjecture: a(n) > 0 for all n > 1.
This is similar to the author's conjecture in A303540.
It has been verified that a(n) > 0 for all n = 2..6*10^8.

			a(9) = 1 with 9 = 1^2 + 2^2 + binomial(2*0+1,0) + binomial(2*1+1,1).
a(2530) = 1 with 2530 = 0^2 + 49^2 + binomial(2*1+1,1) + binomial(2*4+1,4).
a(3258) = 1 with 3258 = 22^2 + 52^2 + binomial(2*3+1,3) + binomial(2*3+1,3).
a(5300) = 1 with 5300 = 10^2 + 59^2 + binomial(2*1+1,1) + binomial(2*6+1,6).
a(13453) = 1 with 13453 = 51^2 + 104^2 + binomial(2*0+1,0) + binomial(2*3+1,3).
a(20964) = 1 with 20964 = 13^2 + 138^2 + binomial(2*3+1,3) + binomial(2*6+1,6).

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, New conjectures on representations of integers (I), Nanjing Univ. J. Math. Biquarterly 34(2017), no. 2, 97-120.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000290, A001481, A001700, A273812, A302982, A302984, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303540, A303541, A303543, A303601.

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n+1,n];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1],4]==3&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=(n==0)||(n>0&&g[n]);
tab={};Do[r=0;k=0;Label[bb];If[c[k]>n,Goto[aa]];Do[If[QQ[n-c[k]-c[j]],Do[If[SQ[n-c[k]-c[j]-x^2],r=r+1],{x,0,Sqrt[(n-c[k]-c[j])/2]}]],{j,0,k}];k=k+1;Goto[bb];Label[aa];tab=Append[tab,r],{n,1,80}];Print[tab]

A000415 Numbers that are the sum of 2 but no fewer nonzero squares.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 26, 29, 32, 34, 37, 40, 41, 45, 50, 52, 53, 58, 61, 65, 68, 72, 73, 74, 80, 82, 85, 89, 90, 97, 98, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 170, 173, 178, 180, 181

Offset: 1

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

Only these numbers can occur as discriminants of quintic polynomials with solvable Galois group F20. - Artur Jasinski, Oct 25 2007
Complement of A022544 in the nonsquare positive integers A000037. - Max Alekseyev, Jan 21 2010
Nonsquare positive integers D such that Pell equation y^2 - D*x^2 = -1 has rational solutions. - Max Alekseyev, Mar 09 2010
Nonsquares for which all 4k+3 primes in the integer's canonical form occur with even multiplicity. - Ant King, Nov 02 2010

E. Grosswald, Representation of Integers as Sums of Squares, Springer-Verlag, New York Inc., (1985), p.15. - Ant King, Nov 02 2010

T. D. Noe, Table of n, a(n) for n = 1..10000
R. K. Guy, Every number is expressible as the sum of how many polygonal numbers?, Amer. Math. Monthly 101 (1994), 169-172. - _Ant King_, Nov 02 2010
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Cf. A000404, A000419, A001481, A002828, A009003, A134422.

c = {}; Do[Do[k = a^2 + b^2; If[IntegerQ[Sqrt[k]], Null, AppendTo[c,k]], {a, 1, 100}], {b, 1, 100}]; Union[c] (* Artur Jasinski, Oct 25 2007 *)
Select[Range[181],Length[PowersRepresentations[ #,2,2]]>0 && !IntegerQ[Sqrt[ # ]] &] (* Ant King, Nov 02 2010 *)

is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); !issquare(n) \\ Charles R Greathouse IV, Feb 07 2017

from itertools import count, islice
from sympy import factorint
def A000415_gen(startvalue=2): # generator of terms >= startvalue
    for n in count(max(startvalue,2)):
        f = factorint(n).items()
        if any(e&1 for p,e in f if p&3<3) and not any(e&1 for p,e in f if p&3==3):
            yield n
A000415_list = list(islice(A000415_gen(),20)) # Chai Wah Wu, Aug 01 2023

{ A000404 } minus { A134422 }. - Artur Jasinski, Oct 25 2007

More terms from Arlin Anderson (starship1(AT)gmail.com)

A300441 Number of the integers 4^k(4u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k(4u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....

Original entry on oeis.org

1, 1, 2, 1, 3, 2, 1, 1, 4, 3, 3, 2, 3, 1, 3, 1, 3, 4, 4, 3, 5, 3, 3, 2, 4, 3, 3, 1, 5, 3, 3, 1, 6, 3, 4, 4, 5, 4, 4, 3, 6, 5, 4, 3, 5, 3, 4, 2, 5, 4, 5, 3, 4, 3, 5, 1, 5, 5, 3, 3, 3, 3, 5, 1, 5, 6, 3, 3, 6, 4, 4, 4, 6, 5, 5, 4, 6, 4, 5, 3

Offset: 1

Zhi-Wei Sun, Mar 05 2018

nonn

Conjecture: a(n) > 0 for all n > 0, and a(n) = 1 only for n = 2^k*m (k = 0,1,2,... and m = 1, 7).
This curious conjecture indicates that any positive square can be written as (2^k)^2 + (2^(k+1)*u(m))^2 + x^2 + y^2 with k,m,x,y nonnegative integers. In the 2017 JNT paper, the author proved that each n = 1,2,3,... can be written as 4^k*(1+4*x^2+y^2)+z^2 with k,x,y,z nonnegative integers.
We have verified that a(n) > 0 for all n = 1..10^7.

			a(1) = 1 since 1^2 - 4^0*(4*u(0)^2+1) = 1 is 1^2 + 0^2.
a(5) = 3 since 5^2 - 4^0*(4*u(1)^2+1) = 20 = 4^2 + 2^2, 5^2 - 4^1*(4*u(1)^2+1) = 5 = 2^2 + 1^2, and 5^2 - 4^2*(4*u(0)^2+1) = 9 = 3^2 + 0^2.
a(7) = 1 since 7^2 - 4^1*(4*u(0)^2+1) = 45 = 6^2 + 3^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000118, A000290, A000302, A001353, A001481, A271518, A281976, A299924, A299537, A299794, A300219, A300362, A300396.

u[0]=0;
u[1]=1;
u[n_]:=u[n]=4u[n-1]-u[n-2];
f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[If[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1,1,0],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
tab={};Do[r=0;Do[m=0;Label[cc];If[4u[m]^2+1>n^2/4^k,Goto[bb]];If[QQ[n^2-4^k*(4u[m]^2+1)],r=r+1,m=m+1;Goto[cc]];
Label[bb],{k,0,Log[2,n]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A020668 Numbers of the form x^2 + 4*y^2.

Original entry on oeis.org

0, 1, 4, 5, 8, 9, 13, 16, 17, 20, 25, 29, 32, 36, 37, 40, 41, 45, 49, 52, 53, 61, 64, 65, 68, 72, 73, 80, 81, 85, 89, 97, 100, 101, 104, 109, 113, 116, 117, 121, 125, 128, 136, 137, 144, 145, 148, 149, 153, 157, 160, 164, 169, 173, 180, 181, 185, 193, 196, 197, 200, 205, 208

Offset: 1

David W. Wilson

x^2 + 4y^2 has discriminant -16.
Numbers that can be expressed as both the sum of two squares and the difference of two squares; the intersection of sequences A001481 and A042965. - T. D. Noe, Feb 05 2003
A004531(n) is nonzero if and only if n is of the form x^2 + 4*y^2. - Michael Somos, Jan 05 2012
These are the sum of two squares that are congruent to 0 or 1 (mod 4), and thus that are also the difference of two squares. - Jean-Christophe Hervé, Oct 25 2015

Jean-Christophe Hervé, Table of n, a(n) for n = 1..7500
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A001481, A004531, A042965, A097269. For primes see A002144.

[n: n in [0..208] | NormEquation(4, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

Select[Range[0, 300], SquaresR[2, #] != 0 && Mod[#, 4] != 2&] (* Jean-François Alcover, May 13 2017 *)

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0, print1(n, ", "))) \\ Altug Alkan, Oct 29 2015

Complement of A097269 in A001481. - Jean-Christophe Hervé, Oct 25 2015

A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

Original entry on oeis.org

1, 2, 4, 7, 8, 9, 14, 16, 17, 18, 23, 25, 28, 31, 32, 34, 36, 41, 46, 47, 49, 50, 56, 62, 63, 64, 68, 71, 72, 73, 79, 81, 82, 89, 92, 94, 97, 98, 100, 103, 112, 113, 119, 121, 124, 126, 127, 128, 136, 137, 142, 144, 146, 151, 153, 158, 161, 162, 164, 167, 169, 175, 178

Offset: 1

N. J. A. Sloane

nonn

x^2 - 2y^2 has discriminant 8. - N. J. A. Sloane, May 30 2014
A positive number n is representable in the form x^2 - 2y^2 iff every prime p == 3 or 5 (mod 8) dividing n occurs to an even power.
Indices of nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s)+Kronecker(m,p)*p^(-2s))^(-1) for m=2 (A035185). [amended by Georg Fischer, Sep 03 2020]
Also positive numbers of the form 2x^2 - y^2. If x^2 - 2y^2 = n, 2(x+y)^2 - (x+2y)^2 = n. - Franklin T. Adams-Watters, Nov 09 2009
Except 2, prime numbers in this sequence have the form p=8k+-1. According to the first comment, prime factors of the forms (8k+-3),(8k+-5) occur in x^2 - 2y^2 in even powers. If x^2 - 2y^2 is a prime number, those powers must be 0. Only factors 8k+-1 remain. Example: 137=8*17+1. - Jerzy R Borysowicz, Nov 04 2015
The product of any two terms of the sequence is a term too. A proof follows from the identity: (a^2-2b^2)(c^2-2d^2) = (2bd+ac)^2 - 2(ad+bc)^2. Example: 127*175 has form x^2-2y^2, with x=9335, y=6600. - Jerzy R Borysowicz, Nov 28 2015
Primitive terms (not a product of earlier terms that are greater than 1 in the sequence) are A055673 except 1. - Charles R Greathouse IV, Sep 10 2016
Positive numbers of the form u^2 + 2uv - v^2. - Thomas Ordowski, Feb 17 2017
For integer numbers z, a, k and z^2+a^2>0, k>=0: z^(4k) + a^4 is in A035251 because z^(4k) + a^4 = (z^(2k) + a^2)^2 - 2(a*z^k)^2. Assume 0^0 = 1. Examples: 3^4 + 1^4 = 82, 3^8+4^4=6817. - Jerzy R Borysowicz, Mar 09 2017
Numbers that are the difference between two legs of a Pythagorean right triangle. - Michael Somos, Apr 02 2017

			The (x,y) pairs, with minimum x, that solve the equation are (1,0), (2,1), (2,0), (3,1), (4,2), (3,0), (4,1), (4,0), (5,2), (6,3), (5,1), (5,0), (6,2), (7,3), (8,4), (6,1), (6,0), (7,2), (8,3), (7,1), (7,0), (10,5), (8,2), ... If the positive number is a perfect square, y=0 yields a trivial solution. - _R. J. Mathar_, Sep 10 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
K. Matthews, Thue's theorem and the diophantine equation x^2-D*y^2=+-N, Math. Comp. 71 (239) (2002) 1281-1286.
K. Matthews, Solving the diophantine equation x^2-D*y^2=N, D>0, (2016).
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998
Sci.math, General Pell equation: x^2 - N*y^2 = D, 1998 [Edited and cached copy]
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Cf. A035185, A042965, A001481, A000047.
Primes: A038873.
Complement of A232531. - Thomas Ordowski and Altug Alkan, Feb 09 2017

filter:= proc(n) local F;
  F:= select(t -> t[1] mod 8 = 3 or t[1] mod 8 = 5, ifactors(n)[2]);
  map(t -> t[2],F)::list(even);
end proc:
select(filter, [$1..1000]); # Robert Israel, Dec 01 2015

Reap[For[n = 1, n < 200, n++, r = Reduce[x^2 - 2 y^2 == n, {x, y}, Integers]; If[r =!= False, Sow[n]]]][[2, 1]] (* Jean-François Alcover, Oct 31 2016 *)

select(x -> x, direuler(p=2,201,1/(1-(kronecker(2,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020

{a(n) = my(m, c); if( n<1, 0, c=0; m=0; while( cMichael Somos, Aug 17 2006 */

is(n)=#bnfisintnorm(bnfinit(z^2-2),n) \\ Ralf Stephan, Oct 14 2013

from itertools import count, islice
from sympy import factorint
def A035251_gen(): # generator of terms
    return filter(lambda n:all(not((2 < p & 7 < 7) and e & 1) for p, e in factorint(n).items()),count(1))
A035251_list = list(islice(A035251_gen(),30)) # Chai Wah Wu, Jun 28 2022

Better description from Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

Original entry on oeis.org

1, 3, 4, 6, 8, 10, 11, 13, 15, 17, 19, 20, 22, 24, 26, 28, 30, 33, 35, 37, 39, 41, 45, 47, 48, 50, 52, 54, 56, 60, 62, 64, 66, 67, 69, 71, 73, 75, 77, 79, 81, 83, 85, 89, 90, 94, 96, 98, 102, 104, 106, 108, 110, 112, 114, 115, 117, 119, 123, 125, 127, 129, 131

Offset: 1

Rajan Murthy and Vale Murthy, Nov 24 2013

nonn

The interval between terms reflects the number of ways a square integer can be partitioned into the sum of two square integers in an ordered pair. As examples, the increase from a(1) to a(2) from 1 to 3 is due to the inclusion of (1,2) and (2,1); and the increase from a(2) to a(3) is due to the inclusion of (2,2). Larger intervals occur when there are more combinations, such as, between a(17) and a(18) when (1,7), (7,1), and (5,5) are included.

			When radius of the circle exceeds 2^(1/2), one square is completely within the circle until the radius reaches 5^(1/2) when three squares are completely within the circle.

Robert G. Wilson v, Table of n, a(n) for n = 1..10000 (terms 1..2623 from Rajan Murthy).
L. Edson Jeffery, Illustration of the first few terms.
Rajan Murthy, Graph of sequence
Rajan Murthy, Graph of intervals
Rajan Murthy, Diagram depicting A(13)

First differences are in A229904.
The first differences must be odd at positions given in A024517 by proof by symmetry as r^2=2*n^2 is on the x=y line.
The radii corresponding to the terms are given by the square roots of A000404.
Cf. A001481, A057961.
Cf. A237707 (3-dimensional analog), A239353 (4-dimensional analog).

(* An empirical solution *) terms = 100; f[r_] := Sum[Floor[Sqrt[r^2 - n^2]], {n, 1, Floor[r]}]; Clear[g]; g[m_] := g[m] = Union[Table[f[Sqrt[s]], {s, 2, m }]][[1 ;; terms]]; g[m = dm = 4*terms]; g[m = m + dm]; While[g[m] != g[m - dm], Print[m]; m = m + dm]; A232499 = g[m]  (* Jean-François Alcover, Mar 06 2014 *)

A057653 Odd numbers of form x^2 + y^2.

Original entry on oeis.org

1, 5, 9, 13, 17, 25, 29, 37, 41, 45, 49, 53, 61, 65, 73, 81, 85, 89, 97, 101, 109, 113, 117, 121, 125, 137, 145, 149, 153, 157, 169, 173, 181, 185, 193, 197, 205, 221, 225, 229, 233, 241, 245, 257, 261, 265, 269, 277, 281, 289, 293, 305, 313, 317, 325, 333, 337

Offset: 1

N. J. A. Sloane, Oct 15 2000

nonn

Numbers with only odd prime factors and such that all prime factors congruent to 3 modulo 4 occur to an even exponent. - Jean-Christophe Hervé, Oct 24 2015
Odd terms of A020668. - Altug Alkan, Nov 19 2015
Also one half of the numbers that are the sum of two odd squares (without multiplicity). See A097269 for twice the numbers. - Wolfdieter Lang, Jan 12 2017

Jean-Christophe Hervé, Table of n, a(n) for n = 1..4000
Joerg Arndt, Plane-filling curves on all uniform grids, arXiv preprint arXiv:1607.02433 [math.CO], 2016.
J. H. Conway, E. M. Rains and N. J. A. Sloane, On the existence of similar sublattices, Canad. J. Math. 51 (1999), 1300-1306 (Abstract, pdf, ps).

Odd members of A001481.
Odd members of A020668.
Complement of A084109 in 4k+1 numbers (A016813).
Cf. A016754 (odd squares), A097269.

readlib(issqr): for n from 1 to 1001 by 2 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d,`,n); break fi; od:od:

fQ[n_] := Length@ Catch@ Do[If[IntegerQ@ Sqrt[n - k^2], Throw[{k, Sqrt[n - k^2]}], Nothing], {k, Floor[Sqrt@ n]^2}] != 0; Select[Range[1, 340, 2], fQ] (* Michael De Vlieger, Nov 13 2015 *)

isok(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1;
for(n=1, 1e3, if(isok(n) && n%2==1, print1(n", "))) \\ Altug Alkan, Nov 13 2015

for(n=0, 1e3, if(if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 4], n)[n]) != 0 && n%2==1, print1(n, ", "))) \\ Altug Alkan, Nov 19 2015

from itertools import count, islice
from sympy import factorint
def A057653_gen(): # generator of terms
    return filter(lambda n:all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items()),count(1,2))
A057653_list = list(islice(A057653_gen(),30)) # Chai Wah Wu, Jun 28 2022

n = odd square * {product of distinct primes == 1 (mod 4)}.

a(n) = A097269(n)/2. - Wolfdieter Lang, Jan 12 2017

More terms from James Sellers, Oct 16 2000

A118882 Numbers which are the sum of two squares in two or more different ways.

Original entry on oeis.org

25, 50, 65, 85, 100, 125, 130, 145, 169, 170, 185, 200, 205, 221, 225, 250, 260, 265, 289, 290, 305, 325, 338, 340, 365, 370, 377, 400, 410, 425, 442, 445, 450, 481, 485, 493, 500, 505, 520, 530, 533, 545, 565, 578, 580, 585, 610, 625, 629, 650, 676, 680

Offset: 1

Franklin T. Adams-Watters, May 03 2006

nonn

Numbers whose prime factorization includes at least two primes (not necessarily distinct) congruent to 1 mod 4 and any prime factor congruent to 3 mod 4 has even multiplicity. Products of two values in A004431.

Squares of distances that are the distance between two points in the square lattice in two or more nontrivially different ways. A quadrilateral with sides a,b,c,d has perpendicular diagonals iff a^2+c^2 = b^2+d^2. This sequence is the sums of the squares of opposite sides of such quadrilaterals, excluding kites (a=b,c=d), but including right triangles (the degenerate case d=0).

			50 = 7^2 + 1^2 = 5^2 + 5^2, so 50 is in the sequence.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index entries for sequences related to sums of squares

Cf. A004431, A009177, A085265.

Cf. A007692, A001481, A022544.

import Data.List (findIndices)
a118882 n = a118882_list !! (n-1)
a118882_list = findIndices (> 1) a000161_list
-- Reinhard Zumkeller, Aug 16 2011

Select[Range[1000], Length[PowersRepresentations[#, 2, 2]] > 1&] (* Jean-François Alcover, Mar 02 2019 *)

from itertools import count, islice
from math import prod
from sympy import factorint
def A118882_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        f = factorint(n)
        if 1>1):
            yield n
A118882_list = list(islice(A118882_gen(),30)) # Chai Wah Wu, Sep 09 2022

A000161(a(n)) > 1. [Reinhard Zumkeller, Aug 16 2011]

A300510 Number of ways to write n^2 as 4^k*(m^2+1) + x^2 + y^2, where m is 1 or 2, and k,x,y are nonnegative integers with x <= y.

Original entry on oeis.org

0, 1, 2, 1, 3, 3, 3, 1, 4, 4, 5, 3, 5, 4, 6, 1, 3, 4, 5, 4, 7, 6, 5, 3, 8, 6, 6, 4, 5, 7, 7, 1, 5, 4, 11, 4, 7, 5, 6, 4, 6, 8, 5, 6, 12, 5, 5, 3, 6, 9, 8, 6, 7, 6, 10, 4, 7, 7, 6, 7, 5, 9, 9, 1, 8, 5, 10, 4, 9, 11, 9, 4, 11, 7, 12, 5, 8, 7, 7, 4

Offset: 1

Zhi-Wei Sun, Mar 07 2018

nonn

Conjecture: a(n) > 0 for all n > 1; in other words, for any integer n > 1 there is a nonnegative integer k such that either n^2 - 2*4^k or n^2 - 5*4^k can be written as the sum of two squares. Moreover, a(n) = 1 only for n = 2^k with k > 0.
This conjecture is stronger than the first conjecture in A300448. We have verified that a(n) > 0 for all n = 2..5*10^7.
Consider positive integers c not divisible by 4 such that for any integer n > 1 there is a nonnegative integer k for which n^2 - 2*4^k or n^2 - c*4^k can be written as the sum of two squares. Our computation for n up to 3*10^7 shows that the only candidates for values of c smaller than 160 are 5, 17, 18, 26, 29, 41, 45, 65, 74, 89, 98, 101, 113, 122, 125, 146, 149, 153. These numbers have the form 9^a*(3*b+2) with a and b nonnegative integers and the p-adic order of 3*b+2 is even for any prime p == 3 (mod 4). For n = 42211965 there is no nonnegative integer k such that n^2 - 2*4^k or n^2 - 162*4^k can be written as the sum of two squares.
Qing-Hu Hou at Tianjin Univ. reported that he had verified a(n) > 0 for n up to 10^9. - Zhi-Wei Sun, Mar 14 2018
Qing-Hu Hou found that 29, 65, 113 should be excluded from the candidates. In fact, for c = 29, 65, 113 there is no nonnegative integer k such that N(c)^2 - 2*4^k or N(c)^2 - c*4^k can be written as the sum of two squares, where N(29) = 51883659, N(65) = 56173837 and N(113) = 65525725. - Zhi-Wei Sun, Mar 23 2018
a(n) > 0 for 1 < n < 6*10^9. - Giovanni Resta, Jun 14 2019

			a(1) = 0 since 1^2 - 4^k*(m^2+1) < 0 for k = 0,1,2,... and m = 1, 2.
a(2) = 1 since 2^2 = 4^0*(1^2+1) + 1^2 + 1^2.
a(3) = 2 since 3^2 = 4^0*(2^2+1) + 0^2 + 2^2 = 4^1*(1^2+1) + 0^2 + 1^2.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000
Zhi-Wei Sun, Refining Lagrange's four-square theorem, J. Number Theory 175(2017), 167-190.
Zhi-Wei Sun, Restricted sums of four squares, arXiv:1701.05868 [math.NT], 2017-2018.

Cf. A000118, A000302, A001481, A271518, A281976, A299924, A299537, A299794, A300362, A300396, A300448, A301452, A301471.

f[n_]:=f[n]=FactorInteger[n];
g[n_]:=g[n]=Sum[Boole[Mod[Part[Part[f[n],i],1]-3,4]==0&&Mod[Part[Part[f[n],i],2],2]==1],{i,1,Length[f[n]]}]==0;
QQ[n_]:=QQ[n]=n==0||(n>0&&g[n]);
SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
tab={};Do[r=0;Do[If[QQ[n^2-4^k*(m^2+1)],Do[If[SQ[n^2-4^k(m^2+1)-x^2],r=r+1],{x,0,Sqrt[(n^2-4^k(m^2+1))/2]}]],{m,1,2},{k,0,Log[4,n^2/(m^2+1)]}];tab=Append[tab,r],{n,1,80}];Print[tab]

A001983 Numbers that are the sum of 2 distinct squares: of form x^2 + y^2 with 0 <= x < y.

Original entry on oeis.org

1, 4, 5, 9, 10, 13, 16, 17, 20, 25, 26, 29, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 73, 74, 80, 81, 82, 85, 89, 90, 97, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160, 164

Offset: 1

N. J. A. Sloane

This sequence lists the values of A000404(n)/2 when A000404(n) is an even number. In other words, sequence lists integers n that are the average of two nonzero squares. - Altug Alkan, May 26 2016

T. D. Noe, Table of n, a(n) for n = 1..1000
G. Xiao, Two squares
Index entries for sequences related to sums of squares

Cf. A000404, subsequence of A001481, A004435 (complement), A025435, A004431.

Union of A000290 and A004431 excluding 0.

a001983 n = a001983_list !! (n-1)
a001983_list = [x | x <- [0..], a025435 x > 0]
-- Reinhard Zumkeller, Dec 20 2013

upto=200;max=Floor[Sqrt[upto]];s=Total/@((Subsets[Range[0,max], {2}])^2);Union[Select[s,#<=upto&]]  (* Harvey P. Dale, Apr 01 2011 *)
selQ[n_] := Select[ PowersRepresentations[n, 2, 2], 0 <= #[[1]] < #[[2]] &] != {}; Select[Range[200], selQ] (* Jean-François Alcover, Oct 03 2013 *)

list(lim)=my(v=List()); for(x=0,sqrtint(lim\4), for(y=x+1, sqrtint(lim\1-x^2), listput(v, x^2+y^2))); Set(v) \\ Charles R Greathouse IV, Feb 07 2017

A025435(a(n)) > 0. - Reinhard Zumkeller, Dec 20 2013

cp's OEIS Frontend

A303639 Number of ways to write n as a^2 + b^2 + binomial(2*c+1,c) + binomial(2*d+1,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

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A000415 Numbers that are the sum of 2 but no fewer nonzero squares.

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A300441 Number of the integers 4^k*(4*u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k*(4*u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....

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A020668 Numbers of the form x^2 + 4*y^2.

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Magma

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A035251 Positive numbers of the form x^2 - 2y^2 with integers x, y.

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Maple

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A232499 Number of unit squares, aligned with a Cartesian grid, completely within the first quadrant of a circle centered at the origin ordered by increasing radius.

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A057653 Odd numbers of form x^2 + y^2.

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A303639 Number of ways to write n as a^2 + b^2 + binomial(2c+1,c) + binomial(2d+1,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

A300441 Number of the integers 4^k(4u(m)^2+1) (k,m = 0,1,2,...) such that n^2 - 4^k(4u(m)^2+1) can be written as the sum of two squares, where u(0) = 0, u(1) = 1, and u(j+1) = 4*u(j) - u(j-1) for j = 1,2,3,....