OFFSET
1,3
COMMENTS
Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.
Closed under multiplication. -David W. Wilson,Dec 20 2004
Also, numbers whose cubes are the sum of 2 squares. -Artur Jasinski,Nov 21 2006 (Cf.A125110.)
Terms are the squares of smallest radii of circles covering (on a square grid) a number of points equal to the terms ofA057961.- Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 16 2007. [Comment corrected byT. D. Noe,Mar 28 2008]
Numbers with more 4k+1 divisors than 4k+3 divisors. If a(n) is a member of this sequence, then so too is any power of a(n). -Ant King,Oct 05 2010
Numbers that are the norms of Gaussian integers. This sequence has unique factorization; the primitive elements areA055025.-Franklin T. Adams-Watters,Nov 25 2011
These are numbers n such that all of n's odd prime factors congruent to 3 modulo 4 occur to an even exponent (Fermat's two-squares theorem). -Jean-Christophe Hervé,May 01 2013
Let's say that an integer n divides a lattice if there exists a sublattice of index n. Example: 2, 4, 5 divide the square lattice. The present sequence without 0 is the sequence of divisors of the square lattice. Say that n is a "prime divisor" if the index-n sublattice is not contained in any other sublattice except the original lattice itself. ThenA055025(norms of Gaussian primes) gives the "prime divisors" of the square lattice. -Jean-Christophe Hervé,May 01 2013
For any i,j > 0 a(i)*a(j) is a member of this sequence, since (a^2 + b^2)*(c^2 + d^2) = (a*c + b*d)^2 + (a*d - b*c)^2. -Boris Putievskiy,May 05 2013
The sequence is closed under multiplication. Primitive elements are inA055025.The sequence can be split into 3 multiplicatively closed subsequences: {0},A004431andA125853.-Jean-Christophe Hervé,Nov 17 2013
Generalizing Jasinski's comment, same as numbers whose odd powers are the sum of 2 squares, by Fermat's two-squares theorem. -Jonathan Sondow,Jan 24 2014
By the 4 squares theorem, every nonnegative integer can be expressed as the sum of two elements of this sequence. -Franklin T. Adams-Watters,Mar 28 2015
There are never more than 3 consecutive terms. Runs of 3 terms start at 0, 8, 16, 72,... (A082982). -Ivan Neretin,Nov 09 2015
Conjecture: barring the 0+2, 0+4, 0+8, 0+16,... sequence, the sum of 2 distinct terms in this sequence is never a power of 2. -J. Lowell,Jan 14 2022
All the areas of squares whose vertices have integer coordinates. -Neeme Vaino,Jun 14 2023
Numbers represented by the definite binary quadratic forms x^2 + 2nxy + (n^2+1)y^2 for any integer n. This sequence contains the even powers of any integer. An odd power of a number appears only if the number itself belongs to the sequence. The equation given in the comment by Boris Putievskiy 2013 is Brahmagupta's identity with n = 1. It proves that any set of numbers of the form a^2 + nb^2 is closed under multiplication. -Klaus Purath,Sep 06 2023
REFERENCES
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
L. Euler, (E388) Vollständige Anleitung zur Algebra, Zweiter Theil, reprinted in: Opera Omnia. Teubner, Leipzig, 1911, Series (1), Vol. 1, p. 417.
S. R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
G. H. Hardy, Ramanujan, pp. 60-63.
P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Expos. Math. 17 (1999), pp. 289-312.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
T. D. Noe,Table of n, a(n) for n = 1..10000
Michael Baake, Uwe Grimm, Dieter Joseph and Przemyslaw Repetowicz,Averaged shelling for quasicrystals,arXiv:math/9907156 [math.MG], 1999.
Henry Bottomley,Illustration of initial terms
Richard T. Bumby,Sums of four squares,in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.
John Butcher,Quadratic residues and sums of two squares
John Butcher,Sums of two squares revisited
Leonhard Euler,Vollständige Anleitung zur Algebra, Zweiter Teil;see alsoDeutsches Textarchiv
Steven R. Finch,Landau-Ramanujan Constant[broken link]
Steven R. Finch,Landau-Ramanujan Constant[From the Wayback Machine]
Steven R. Finch,On a Generalized Fermat-Wiles Equation[broken link]
Steven R. Finch,On a Generalized Fermat-Wiles Equation[From the Wayback Machine]
J. W. L. Glaisher,On the function which denotes the difference between the number of (4m+1)-divisors and the number of (4m+3)-divisors of a number,Proc. London Math. Soc., 15 (1884), 104-122. [Annotated scanned copy of pages 104-107 only]
Leonor Godinho, Nicholas Lindsay, and Silvia Sabatini,On a symplectic generalization of a Hirzebruch problem,arXiv:2403.00949 [math.SG], 2024. See p. 17.
Darij Grinberg,UMN Spring 2019 Math 4281 notes,University of Minnesota, College of Science & Engineering, 2019. [Wayback Machine copy]
Shuo Li,The characteristic sequence of the integers that are the sum of two squares is not morphic,arXiv:2404.08822 [math.NT], 2024.
Thomas Nickson and Igor Potapov,Broadcasting Automata and Patterns on Z^2,arXiv preprint arXiv:1410.0573 [cs.FL], 2014.
Michael Penn,Sums of Squares,Youtube playlist, 2019, 2020.
Peter Shiu,Counting Sums of Two Squares: The Meissel-Lehmer Method,Mathematics of Computation 47 (1986), 351-360.
N. J. A. Sloane et al.,Binary Quadratic Forms and OEIS(Index to related sequences, programs, references)
William A. Stein,Quadratic Forms:Sums of Two Squares
Eric Weisstein's World of Mathematics,Square Number
Eric Weisstein's World of Mathematics,Generalized Fermat Equation
Eric Weisstein's World of Mathematics,Landau-Ramanujan Constant
Eric Weisstein's World of Mathematics,Gaussian Integer
A. van Wijngaarden,A table of partitions into two squares with an application to rational triangles,Proceedings of the Koninklijke Nederlandse Akademie van Wetenschappen, Series A, 53 (1950), 869-875.
Gang Xiao,Two squares
FORMULA
n = square * 2^{0 or 1} * {product of distinct primes == 1 (mod 4)}.
The number of integers less than N that are sums of two squares is asymptotic to constant*N/sqrt(log(N)), hence lim_{n->infinity} a(n)/n = infinity.
Nonzero terms in expansion of Dirichlet series Product_p (1 - (Kronecker(m, p) + 1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -1.
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533.-Charles R Greathouse IV,Apr 16 2012
There are B(x) = x/sqrt(log x) * (K + B2/log x + O(1/log^2 x)) terms of this sequence up to x, where K =A064533and B2 =A227158.-Charles R Greathouse IV,Nov 18 2022
MAPLE
readlib(issqr): for n from 0 to 160 do for k from 0 to floor(sqrt(n)) do if issqr(n-k^2) then printf(`%d, `, n); break fi: od: od:
MATHEMATICA
upTo = 160; With[{max = Ceiling[Sqrt[upTo]]}, Select[Union[Total /@ (Tuples[Range[0, max], {2}]^2)], # <= upTo &]] (*Harvey P. Dale,Apr 22 2011 *)
Select[Range[0, 160], SquaresR[2, #]!= 0 &] (*Jean-François Alcover,Jan 04 2013 *)
PROG
(PARI) isA001481(n)=local(x, r); x=0; r=0; while(x<=sqrt(n) && r==0, if(issquare(n-x^2), r=1); x++); r \\Michael B. Porter,Oct 31 2009
(PARI) is(n)=my(f=factor(n)); for(i=1, #f[, 1], if(f[i, 2]%2 && f[i, 1]%4==3, return(0))); 1 \\Charles R Greathouse IV,Aug 24 2012
(PARI) B=bnfinit('z^2+1, 1);
is(n)=#bnfisintnorm(B, n) \\Ralf Stephan,Oct 18 2013, edited byM. F. Hasler,Nov 21 2017
(PARI) list(lim)=my(v=List(), t); for(m=0, sqrtint(lim\=1), t=m^2; for(n=0, min(sqrtint(lim-t), m), listput(v, t+n^2))); Set(v) \\Charles R Greathouse IV,Jan 05 2016
(PARI) is_A001481(n)=!for(i=2-bittest(n, 0), #n=factor(n)~, bittest(n[1, i], 1)&&bittest(n[2, i], 0)&&return) \\M. F. Hasler,Nov 20 2017
(Haskell)
a001481 n = a001481_list!! (n-1)
a001481_list = [x | x <- [0..], a000161 x > 0]
--Reinhard Zumkeller,Feb 14 2012, Aug 16 2011
(Magma) [n: n in [0..160] | NormEquation(1, n) eq true]; //Arkadiusz Wesolowski,May 11 2016
(Python)
from itertools import count, islice
from sympy import factorint
defA001481_gen(): # generator of terms
return filter(lambda n:(lambda m:all(d & 3!= 3 or m[d] & 1 == 0 for d in m))(factorint(n)), count(0))
KEYWORD
nonn,nice,easy,core
AUTHOR
EXTENSIONS
Deleted an incorrect comment. -N. J. A. Sloane,Oct 03 2023
STATUS
approved