A227158 -id:A227158 - cp's OEIS frontend

A001481 Numbers that are the sum of 2 squares.

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

Offset: 1

N. J. A. Sloane

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 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. This sequence has unique factorization; the primitive elements are A055025. - 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
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

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).

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.
Chris Bispels, Matthew Cohen, Joshua Harrington, Joshua Lowrance, Kaelyn Pontes, Leif Schaumann, and Tony W. H. Wong, A further investigation on covering systems with odd moduli, arXiv:2507.16135 [math.NT], 2025. See p. 3.
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 also Deutsches 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
Index entries for sequences related to sums of squares
Index entries for "core" sequences

Disjoint union of A000290 and A000415.
Complement of A022544.
Cf. A004018, A000161, A002654, A064533, A055025, A002828, A000378, A025284-A025320, A125110, A118882, A125022.
A000404 gives another version. Subsequence of A091072, supersequence of A046711.
Cf. A057961, A232499, A077773, A363763.
Column k=2 of A336820.

a001481 n = a001481_list !! (n-1)
a001481_list = [x | x <- [0..], a000161 x > 0]
-- Reinhard Zumkeller, Feb 14 2012, Aug 16 2011

[n: n in [0..160] | NormEquation(1, n) eq true]; // Arkadiusz Wesolowski, May 11 2016

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:

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 *)

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

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

B=bnfinit('z^2+1,1);
is(n)=#bnfisintnorm(B,n) \\ Ralf Stephan, Oct 18 2013, edited by M. F. Hasler, Nov 21 2017

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

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

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))
A001481_list = list(islice(A001481_gen(),30)) # Chai Wah Wu, Jun 27 2022

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 = A064533 and B2 = A227158. - Charles R Greathouse IV, Nov 18 2022

Deleted an incorrect comment. - N. J. A. Sloane, Oct 03 2023

A000404 Numbers that are the sum of 2 nonzero squares.

Original entry on oeis.org

2, 5, 8, 10, 13, 17, 18, 20, 25, 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, 100, 101, 104, 106, 109, 113, 116, 117, 122, 125, 128, 130, 136, 137, 145, 146, 148, 149, 153, 157, 160, 162, 164, 169, 170, 173, 178

Offset: 1

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

From the formula it is easy to see that if k is in this sequence, then so are all odd powers of k. - T. D. Noe, Jan 13 2009
Also numbers whose cubes are the sum of two nonzero squares. - Joe Namnath and Lawrence Sze
A line perpendicular to y=mx has its first integral y-intercept at a^2+b^2. The remaining ones for that slope are multiples of that primitive value. - Larry J Zimmermann, Aug 19 2010
The primes in this sequence are sequence A002313.
Complement of A018825; A025426(a(n)) > 0; A063725(a(n)) > 0. - Reinhard Zumkeller, Aug 16 2011
If the two squares are not equal, then any power is still in the sequence: if k = x^2 + y^2 with x != y, then k^2 = (x^2-y^2)^2 + (2xy)^2 and k^3 = (x(x^2-3y^2))^2 + (y(3x^2-y^2))^2, etc. - Carmine Suriano, Jul 13 2012
There are never more than 3 consecutive terms that differ by 1. Triples of consecutive terms that differ by 1 occur infinitely many times, for example, 2(k^2 + k)^2, (k^2 - 1)^2 + (k^2 + 2 k)^2, and (k^2 + k - 1)^2 + (k^2 + k + 1)^2 for any integer k > 1. - Ivan Neretin, Mar 16 2017 [Corrected by Jerzy R Borysowicz, Apr 14 2017]
Number of terms less than 10^k, k=1,2,3,...: 3, 34, 308, 2690, 23873, 215907, 1984228, ... - Muniru A Asiru, Feb 01 2018
The squares in this sequence are the squares of the so-called hypotenuse numbers A009003. - M. F. Hasler, Jun 20 2025

			25 = 3^2 + 4^2, therefore 25 is a term. Note that also 25^3 = 15625 = 44^2 + 117^2, therefore 15625 is a term.

David A. Cox, "Primes of the Form x^2 + n y^2", Wiley, 1989.
GCHQ, The GCHQ Puzzle Book, Penguin, 2016. See page 103.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 75, Theorem 4, with Theorem 2, p. 15.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 251, 252.
Ian Stewart, "Game, Set and Math", Chapter 8, 'Close Encounters of the Fermat Kind', Penguin Books, Ed. 1991, pp. 107-124.

T. D. Noe, Table of n, a(n) for n = 1..10000
J. M. De Koninck and V. Ouellet, On the n-th element of a set of positive integers, Annales Univ. Sci. Budapest Sect. Comput. 44 (2015), 153-164. See 2. on p. 162.
Étienne Fouvry, Claude Levesque, and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing integers as the sum of two squares in the ring Z_n, arXiv:1404.0187 [math.NT], 2014.
David Rabahy, Google Sheets
G. Xiao, Two squares.
Reinhard Zumkeller, Illustration for A084888 and A000404.
Index entries for sequences related to sums of squares

A001481 gives another version (allowing for zero squares).
Cf. A004431 (2 distinct squares), A063725 (number of representations), A024509 (numbers with multiplicity), A025284, A018825. Also A050803, A050801, A001105, A033431, A084888, A000578, A000290, A057961, A232499, A007692.
Cf. A003325 (analog for cubes), A003336 (analog for 4th powers).
Cf. A009003 (square roots of the squares in this sequence).
Column k=2 of A336725.
Cf. A355237, A355238.

P:=List([1..10^4],i->i^2);;
A000404 := Set(Flat(List(P, i->List(P, j -> i+j)))); # Muniru A Asiru, Feb 01 2018

import Data.List (findIndices)
a000404 n = a000404_list !! (n-1)
a000404_list = findIndices (> 0) a025426_list
-- Reinhard Zumkeller, Aug 16 2011

lst:=[]; for n in [1..178] do f:=Factorization(n); if IsSquare(n) then for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 1 then Append(~lst, n); break; end if; end for; else t:=0; for m in [1..#f] do d:=f[m]; if d[1] mod 4 eq 3 and d[2] mod 2 eq 1 then t:=1; break; end if; end for; if t eq 0 then Append(~lst, n); end if; end if; end for; lst; // Arkadiusz Wesolowski, Feb 16 2017

nMax:=178: A:={}: for i to floor(sqrt(nMax)) do for j to floor(sqrt(nMax)) do if i^2+j^2 <= nMax then A := `union`(A, {i^2+j^2}) else  end if end do end do: A; # Emeric Deutsch, Jan 02 2017

nMax=1000; n2=Floor[Sqrt[nMax-1]]; Union[Flatten[Table[a^2+b^2, {a,n2}, {b,a,Floor[Sqrt[nMax-a^2]]}]]]
Select[Range@ 200, Length[PowersRepresentations[#, 2, 2] /. {0, } -> Nothing] > 0 &] (* _Michael De Vlieger, Mar 24 2016 *)
Module[{upto=200},Select[Union[Total/@Tuples[Range[Sqrt[upto]]^2,2]],#<= upto&]] (* Harvey P. Dale, Sep 18 2021 *)

is_A000404(n)= for( i=1,#n=factor(n)~%4, n[1,i]==3 && n[2,i]%2 && return); n && ( vecmin(n[1,])==1 || (n[1,1]==2 && n[2,1]%2)) \\ M. F. Hasler, Feb 07 2009

list(lim)=my(v=List(),x2); lim\=1; for(x=1,sqrtint(lim-1), x2=x^2; for(y=1,sqrtint(lim-x2), listput(v,x2+y^2))); Set(v) \\ Charles R Greathouse IV, Apr 30 2016

from itertools import count, islice
from sympy import factorint
def A000404_gen(startvalue=1): # generator of terms >= startvalue
    for n in count(max(startvalue,1)):
        c = False
        for p in (f:=factorint(n)):
            if (q:= p & 3)==3 and f[p]&1:
                break
            elif q == 1:
                c = True
        else:
            if c or f.get(2,0)&1:
                yield n
A000404_list = list(islice(A000404_gen(),30)) # Chai Wah Wu, Jul 01 2022

Let k = 2^t * p_1^a_1 * p_2^a_2 * ... * p_r^a_r * q_1^b_1 * q_2^b_2 * ... * q_s^b_s with t >= 0, a_i >= 0 for i=1..r, where p_i == 1 (mod 4) for i=1..r and q_j == -1 (mod 4) for j=1..s. Then k is a term iff 1) b_j == 0 (mod 2) for j=1..s and 2) r > 0 or t == 1 (mod 2) (or both).
From Charles R Greathouse IV, Nov 18 2022: (Start)
a(n) ~ k*n*sqrt(log n), where k = 1.3085... = 1/A064533.
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 = A064533 and B2 = A227158. (End)

Edited by Ralf Stephan, Nov 15 2004
Typo in formula corrected by M. F. Hasler, Feb 07 2009
Erroneous Mathematica program fixed by T. D. Noe, Aug 07 2009
PARI code fixed for versions > 2.5 by M. F. Hasler, Jan 01 2013

A064533 Decimal expansion of Landau-Ramanujan constant.

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 08 2001

Named after the German mathematician Edmund Georg Hermann Landau (1877-1938) and the Indian mathematician Srinivasa Ramanujan (1887-1920). - Amiram Eldar, Jun 20 2021

			0.76422365358922066299069873125009232811679054139340951472168667374...

Bruce C. Berndt, Ramanujan's notebook part IV, Springer-Verlag, 1994, pp. 52, 60-66; MR 95e: 11028.
Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.
G. H. Hardy, "Ramanujan, Twelve lectures on subjects suggested by his life and work", Chelsea, 1940, pp. 60-63; MR 21 # 4881.
Edmund Landau, Über die Einteilung der positiven ganzen Zahlen in vier Klassen nach der Mindestzahl der zu ihrer additiven Zusammensetzung erforderlichen Quadrate. Arch. Math. Phys., 13, 1908, pp. 305-312.

David E. G. Hare, Table of n, a(n) for n = 0..125078
Bruce C. Berndt and Pieter Moree, Sums of two squares and the tau-function: Ramanujan's trail, arXiv:2409.03428 [math.NT], 2024. See p. 30.
Alexandru Ciolan, Alessandro Languasco and Pieter Moree, Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms, section 10, Journal of Mathematical Analysis and Applications, 2022; see also preprint on arXiv, arXiv:2109.03288 [math.NT], 2021.
Alessandro Languasco, Programs and numerical results, providing 130000 digits. [Note: information ancillary to above link.]
Steven R. Finch, Landau-Ramanujan Constant. [Broken link]
Steven R. Finch, Landau-Ramanujan Constant. [From the Wayback machine]
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]
Philippe Flajolet and Ilan Vardi, Zeta function expansions of some classical constants, Feb 18 1996.
Étienne Fouvry, Claude Levesque and Michel Waldschmidt, Representation of integers by cyclotomic binary forms, arXiv:1712.09019 [math.NT], 2017.
Xavier Gourdon and Pascal Sebah, Constants and records of computation.
David E. G. Hare, 125,079 digits of the Landau-Ramanujan constant.
David E. G. Hare, Landau-Ramanujan constant up to 10000 digits.
Institute of Physics, Constants - Landau-Ramanujan Constant.
Simon Plouffe, Landau Ramanujan constant.
Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18, No. 85 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Ramanujan constant.
Wikipedia, Landau-Ramanujan constant.
Robert G. Wilson v, The first 15584 digits of the Landau-Ramanujan constant.

Cf. A125776 = Continued fraction. - Harry J. Smith, May 13 2009

Cf. A000692, A001481, A009003, A075880, A090735, A090736, A227158.

First@ RealDigits@ N[1/Sqrt@2 Product[((1 - 2^(-2^k)) 4^(2^k) Zeta[2^k]/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^(2^(-k - 1)), {k, 8}], 2^8] (* Robert G. Wilson v, Jul 01 2007 *)
(* Victor Adamchik calculated 5100 digits of the Landau-Ramanujan constant using Mathematica (from Mathematica 4 demos): *) LandauRamanujan[n_] := With[{K = Ceiling[Log[2, n*Log[3, 10]]]}, N[Product[(((1 - 2^(-2^k))*4^2^k*Zeta[2^k])/(Zeta[2^k, 1/4] - Zeta[2^k, 3/4]))^2^(-k - 1), {k, 1, K}]/Sqrt[2], n]];
(* The code reported here is the code at https://library.wolfram.com/infocenter/Demos/120/. Looking carefully at the outputs reported there one sees that: the last 8 digits of the 500-digit output ("74259724") are not the same as those listed in the 1000-digit output ("94247095"); the same happens with the last 18 digits of the 1000-digit output ("584868265713856413") and the corresponding ones in the 5100-digit output ("852514327407923660"). - Alessandro Languasco, May 07 2021 *)

From Amiram Eldar, Mar 08 2024: (Start)
Equals (Pi/4) * Product_{primes p == 1 (mod 4)} (1 - 1/p^2)^(1/2).
Equals (1/sqrt(2)) * Product_{primes p == 3 (mod 4)} (1 - 1/p^2)^(-1/2).
Equals (1/sqrt(2)) * Product_{k>=1} ((1 - 1/2^(2^k)) * zeta(2^k)/beta(2^k)), where beta is the Dirichlet beta function (Shanks, 1964). (End)

More references needed! Hardy and Wright? Gruber and Lekkerkerker?

More terms from Vladeta Jovovic, Oct 08 2001

A102548 Number of positive integers <= n that are expressible in the form u^2+v^2, with u and v integers.

Original entry on oeis.org

1, 2, 2, 3, 4, 4, 4, 5, 6, 7, 7, 7, 8, 8, 8, 9, 10, 11, 11, 12, 12, 12, 12, 12, 13, 14, 14, 14, 15, 15, 15, 16, 16, 17, 17, 18, 19, 19, 19, 20, 21, 21, 21, 21, 22, 22, 22, 22, 23, 24, 24, 25, 26, 26, 26, 26, 26, 27, 27, 27, 28, 28, 28, 29, 30, 30, 30, 31, 31, 31, 31, 32, 33, 34, 34

Offset: 1

Salvador Perez Gomez (pies314(AT)hotmail.com), Feb 24 2005

			a(8) = 5 because 1 = 0^2 + 1^2, 2 = 1^2 + 1^2, 4 = 0^2 + 2^2, 5 = 1^2 + 2^2, 8 = 2^2 + 2^2, but 3,6 and 7 are not of the form u^2 + v^2, with u and v integers.

David A. Corneth, Table of n, a(n) for n = 1..10000
Yoichi Motohashi, On the distribution of prime numbers which are of the form x^2+y^2+1, Acta Arith. 16 (1969/70), 351-363. MR0288086 (44 #5284). See Eq. (1.2).
D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.

Cf. A000161, A000691, A001481, A229062.

a := proc(n) local aux,i,m,u,v; aux:=0; for i from 1 to n do m:=floor(sqrt(i/2)); for u from 0 to m do v:=sqrt(i-u^2); if (v = floor(v)) then aux:=aux+1; u:=m; end if; end do; end do; aux; end proc:

a[1]=1; a[n_]:= a[n]= a[n-1] + If[SquaresR[2, n]>0, 1, 0]; Table[a[n], {n,75}] (* Jean-François Alcover, Mar 31 2015 *)

first(n)= my(v = vector(n + 1), res = vector(n)); res[1] = 1; for(i = 0, sqrtint(n), for(j = i, sqrtint(n - i^2), v[i^2+j^2+1] = 1 ) ); for(i = 2, #res, res[i] = res[i-1] + v[i+1]; ); res \\ David A. Corneth, Jun 05 2020

from itertools import count, accumulate, islice
from sympy import factorint
def A102548_gen(): # generator of terms
    return accumulate(int(all(p & 3 != 3 or e & 1 == 0 for p, e in factorint(n).items())) for n in count(1))
A102548_list = list(islice(A102548_gen(),30)) # Chai Wah Wu, Jun 28 2022

From David A. Corneth, Jun 05 2020: (Start)
A000161(a(n)) > 0.
a(n) = (partial sum of A229062 up to n) - 1. (End)
a(n) = n/sqrt(log n) * (K + B2/log n + O(1/log^2 n)), where K = A064533 and B2 = A227158. In particular, a(n) ~ Kn/sqrt(log n). - Charles R Greathouse IV, Dec 03 2022

Name clarified by David A. Corneth, Jun 05 2020

A000692 An approximation to population of x^2 + y^2 <= 2^n.

Original entry on oeis.org

1, 3, 4, 5, 9, 15, 27, 50, 92, 171, 322, 610, 1161, 2220, 4260, 8201, 15828, 30622, 59362, 115287, 224260, 436871, 852161, 1664196, 3253531, 6366973, 12471056, 24447507, 47962236, 94161474, 184983976, 363632192, 715220838, 1407510311

Offset: 0

N. J. A. Sloane

nonn

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).

D. Hare, The constant c [Dave Hare, May 21 1996].
D. Shanks, The second-order term in the asymptotic expansion of B(x), Math. Comp., 18 (1964), 75-86.
Index entries for sequences related to populations of quadratic forms.

Cf. A064533.

Other population sequences for x^2 + y^2: A000050, A000690, A000691.

a(n) = (b*2^n / sqrt(n*log(2))) * (1 + c/(n*log(2))) where b=0.764223654... is the Landau-Ramanujan constant (A064533) and c=0.5819486593... is the second-order Landau-Ramanujan constant (A227158) given by c = (1/2) * (1-log(Pi*e^gamma/(2*L))) - (1/4) * D(1) where D(s) = (d/ds)(log(Product_{p prime == 3 (mod 4)} 1/(1-p^(-2*s)))) and L is the Lemniscate constant (A064853) [see (12) in Shanks]. - Sean A. Irvine, Feb 25 2011

More terms from Sean A. Irvine, Feb 24 2011

Name clarified by Seth A. Troisi, May 23 2022

A164775 a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

Original entry on oeis.org

7, 43, 330, 2749, 24028, 216341, 1985459, 18457847, 173229058, 1637624156, 15570512744, 148736628858, 1426306930865, 13722217893214, 132387263219058, 1280309691127436

Offset: 1

Eric W. Weisstein, Aug 26 2009

			a(1)=7 since 1 = 0^2 + 1^2, 2 = 1^2 + 1^2, 4 = 0^2 + 2^2, 5 = 1^2 + 2^2, 8 = 2^2 + 2^2, 9 = 0^2 + 3^2, 10 = 1^2 + 3^3.

Peter Shiu, Counting Sums of Two Squares: The Meissel-Lehmer Method, Mathematics of Computation 47:175 (July 1986), pp. 351-360. [Beware errors.]
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant

Cf. A000050, A001481, A064533, A227158, A275649, A275650.

a(n) = A180416(n) + ceiling(sqrt(10^n)). - Hiroaki Yamanouchi, Jul 14 2014

Offset changed from 0 to 1 by Robert G. Wilson v, Aug 29 2009
a(9) from Eric W. Weisstein, Aug 29 2009
a(10) from Donovan Johnson, Sep 16 2009
a(11)-a(12) from Ant King, May 02 2010
a(11)-a(12) corrected and a(13)-a(16) added by Hiroaki Yamanouchi, Jul 14 2014

A244662 Decimal expansion of 'C' (as designated by D. Shanks), a constant appearing in the second order term of the asymptotic expansion of the number of non-hypotenuse numbers not exceeding a given bound.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Jul 04 2014

			0.70475345170594788412255819759189881852...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, 2.3 Landau-Ramanujan Constant, p. 101.

Daniel Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation, Vol. 18 (1964), pp. 75-86.
Daniel Shanks, Non-hypotenuse Numbers, Fib. Quart., 13:4 (1975), pp. 319-321.
Eric Weisstein's MathWorld, Lemniscate Constant

Cf. A009003, A004144, A062539, A227158, A244659 (first order term).

digits = 100; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; c = A227158 = f[m]; c + 1/2 Log[(Pi/L)^2*Exp[EulerGamma]/2] // RealDigits[#, 10, digits] & // First

C = c + 1/2*log((Pi/L)^2*exp(gamma)/2), where c is A227158 and L the Lemniscate constant A062539.

A242013 Decimal expansion of the Euler-Kronecker constant (as named by P. Moree) for hypotenuse numbers.

Original entry on oeis.org

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

Offset: 0

Jean-François Alcover, Aug 11 2014

130000 digits are available (see Links). - Alessandro Languasco, Mar 27 2024

			-0.1638973186345815958562997690047351186096657461435450436468425985305...

Steven R. Finch, Mathematical Constants, Cambridge University Press, 2003, Section 2.3 Landau-Ramanujan constants, p. 99.

Steven R. Finch, Errata and Addenda to Mathematical Constants, arXiv:2001.00578 [math.HO], 202-2024, p. 11.
Alessandro Languasco, Programs and numerical results for the paper "Landau and Ramanujan approximations for divisor sums and coefficients of cusp forms".
Alessandro Languasco, Shanks' asymptotic constants for the number of positive integers less or equal than x that are the sum of two squares (Source code), gp script, 2024.
Pieter Moree, Counting numbers in multiplicative sets: Landau versus Ramanujan, p. 13, arXiv:1110.0708v1 [math.NT] 4 Oct 2011.
D. Shanks, The second-order term in the asymptotic expansion of B(x), Mathematics of Computation 18 (1964), pp. 75-86.
Eric Weisstein's World of Mathematics, Landau-Ramanujan Constant.

Cf. A227158.

digits = 101; m0 = 5; dm = 5; beta[x_] := 1/4^x*(Zeta[x, 1/4] - Zeta[x, 3/4]); L = Pi^(3/2)/Gamma[3/4]^2*2^(1/2)/2; Clear[f]; f[m_] := f[m] = 1/2*(1 - Log[Pi*E^EulerGamma/(2*L)]) - 1/4*NSum[Zeta'[2^k]/Zeta[2^k] - beta'[2^k]/beta[2^k] + Log[2]/(2^(2^k) - 1), {k, 1, m}, WorkingPrecision -> digits + 10]; f[m0]; f[m = m0 + dm]; While[RealDigits[f[m], 10, digits] != RealDigits[f[m - dm], 10, digits], m = m + dm]; RealDigits[1 - 2*f[m], 10, digits] // First

1 - 2*A227158.

cp's OEIS Frontend

A001481 Numbers that are the sum of 2 squares.

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A000404 Numbers that are the sum of 2 nonzero squares.

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A064533 Decimal expansion of Landau-Ramanujan constant.

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A102548 Number of positive integers <= n that are expressible in the form u^2+v^2, with u and v integers.

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A000692 An approximation to population of x^2 + y^2 <= 2^n.

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A164775 a(n) is the number of positive integers <= 10^n that can be expressed as a sum of two squares.

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