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A001481 Numbers that are the sum of 2 squares.
(Formerly M0968 N0361)
111
0, 1, 2, 4, 5, 8, 9, 10, 13, 16, 17, 18, 20, 25, 26, 29, 32, 34, 36, 37, 40, 41, 45, 49, 50, 52, 53, 58, 61, 64, 65, 68, 72, 73, 74, 80, 81, 82, 85, 89, 90, 97, 98, 100, 101, 104, 106, 109, 113, 116, 117, 121, 122, 125, 128, 130, 136, 137, 144, 145, 146, 148, 149, 153, 157, 160 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.

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 of A057961. - Philippe Lallouet (philip.lallouet(AT)wanadoo.fr), Apr 16 2007. [Comment corrected by T. 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

A000161(a(n)) > 0; A070176(a(n)) = 0. - Reinhard Zumkeller, Feb 04 2012, Aug 16 2011

Numbers that are the norms of Gaussian integers. - 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. Then A055025 (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 in A055025. The sequence can be split into 3 multiplicatively closed subsequences: {0}, A004431 and A125853. - 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

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) Vollstaendige 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).

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.

LINKS

T. D. Noe, Table of n, a(n) for n = 1..10000

M. Baake, U. Grimm, D. Joseph and P. Repetowicz, Averaged shelling for quasicrystals

H. Bottomley, Illustration of initial terms

R. T. Bumby, Sums of four squares, in Number theory (New York, 1991-1995), 1-8, Springer, New York, 1996.

J. Butcher, Quadratic residues and sums of two squares

J. Butcher, Sums of two squares revisited

L. Euler, Vollstaendige Anleitung zur Algebra, Zweiter Teil.

S. R. Finch, Landau-Ramanujan Constant

S. R. Finch, On a Generalized Fermat-Wiles Equation

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)

W. 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

G. Xiao, Two squares

Index entries for sequences related to sums of squares

Index entries for "core" sequences

FORMULA

n = square * 2^{0 or 1} * {product of distinct primes == 1 (mod 4)}.

The number of integers <N that are sums of two squares is asymptotic to constant*N/sqrt(log(N)), hence lim n->inf a(n)/n = infinity.

Closed under multiplication. - David W. Wilson, Dec 20 2004

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.

Sequence has unique factorization; the primitive elements are A055025. - Franklin T. Adams-Watters, Nov 25 2011

a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533. - Charles R Greathouse IV, Apr 16 2012

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) is(n)=#bnfisintnorm(bnfinit(z^2+1), n) \\ Ralf Stephan, Oct 18 2013

(Haskell)

a001481 n = a001481_list !! (n-1)

a001481_list = [x | x <- [0..], a000161 x > 0]

-- Reinhard Zumkeller, Feb 14 2012, Aug 16 2011

CROSSREFS

Union of A000290 and A000404. Complement of A022544.

Cf. A004018, A000161, A002654, A064533, A055025, A002828, A000378, A025284-A025320, A125110, A118882, A125022.

A000404 gives another version. Subsequence of A091072 and of A046711.

Cf. A057961, A232499.

Sequence in context: A084581 A121996 A091072 * A245226 A034026 A125022

Adjacent sequences:  A001478 A001479 A001480 * A001482 A001483 A001484

KEYWORD

nonn,nice,easy,core,changed

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified September 23 00:13 EDT 2014. Contains 247086 sequences.