

A001481


Numbers that are the sum of 2 squares.
(Formerly M0968 N0361)


228



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
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OFFSET

1,3


COMMENTS

Numbers n such that n = x^2 + y^2 has a solution in nonnegative integers x, y.
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
Numbers that are the norms of Gaussian integers. This sequence has unique factorization; the primitive elements are A055025.  Franklin T. AdamsWatters, 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 twosquares theorem).  JeanChristophe 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 indexn 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.  JeanChristophe 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.  JeanChristophe Hervé, Nov 17 2013
Generalizing Jasinski's comment, same as numbers whose odd powers are the sum of 2 squares, by Fermat's twosquares 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. AdamsWatters, 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", SpringerVerlag, 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. 98104.
G. H. Hardy, Ramanujan, pp. 6063.
P. Moree and J. Cazaran, On a claim of Ramanujan in his first letter to Hardy, Expos. Math. 17 (1999), pp. 289312.
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

Richard T. Bumby, Sums of four squares, in Number theory (New York, 19911995), 18, Springer, New York, 1996.


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.


MAPLE

readlib(issqr): for n from 0 to 160 do for k from 0 to floor(sqrt(n)) do if issqr(nk^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 *)


PROG

(PARI) isA001481(n)=local(x, r); x=0; r=0; while(x<=sqrt(n) && r==0, if(issquare(nx^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);
(PARI) list(lim)=my(v=List(), t); for(m=0, sqrtint(lim\=1), t=m^2; for(n=0, min(sqrtint(limt), m), listput(v, t+n^2))); Set(v) \\ Charles R Greathouse IV, Jan 05 2016
(PARI) is_A001481(n)=!for(i=2bittest(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 !! (n1)
a001481_list = [x  x < [0..], a000161 x > 0]
(Python)
from itertools import count, islice
from sympy import factorint
def A001481_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))


CROSSREFS

Cf. A004018, A000161, A002654, A064533, A055025, A002828, A000378, A025284A025320, A125110, A118882, A125022.


KEYWORD

nonn,nice,easy,core


AUTHOR



EXTENSIONS



STATUS

approved



