A001481 -id:A001481 - cp's OEIS frontend

A000161 Number of partitions of n into 2 squares.

1, 1, 1, 0, 1, 1, 0, 0, 1, 1, 1, 0, 0, 1, 0, 0, 1, 1, 1, 0, 1, 0, 0, 0, 0, 2, 1, 0, 0, 1, 0, 0, 1, 0, 1, 0, 1, 1, 0, 0, 1, 1, 0, 0, 0, 1, 0, 0, 0, 1, 2, 0, 1, 1, 0, 0, 0, 0, 1, 0, 0, 1, 0, 0, 1, 2, 0, 0, 1, 0, 0, 0, 1, 1, 1, 0, 0, 0, 0, 0, 1, 1, 1, 0, 0, 2, 0, 0, 0, 1, 1, 0, 0, 0, 0, 0, 0, 1, 1, 0, 2, 1, 0, 0, 1, 0, 1, 0

Offset: 0

N. J. A. Sloane

Number of ways of writing n as a sum of 2 (possibly zero) squares when order does not matter.
Number of similar sublattices of square lattice with index n.
Let Pk = the number of partitions of n into k nonzero squares. Then we have A000161 = P0 + P1 + P2, A002635 = P0 + P1 + P2 + P3 + P4, A010052 = P1, A025426 = P2, A025427 = P3, A025428 = P4. - Charles R Greathouse IV, Mar 08 2010, amended by M. F. Hasler, Jan 25 2013
a(A022544(n))=0; a(A001481(n))>0; a(A125022(n))=1; a(A118882(n))>1. - Reinhard Zumkeller, Aug 16 2011

			25 = 3^2+4^2 = 5^2, so a(25) = 2.

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 339

T. D. Noe, Table of n, a(n) for n = 0..10000
B. K. Agarwala and F. C. Auluck, Statistical mechanics and partitions into non-integral powers of integers, Proc. Camb. Phil. Soc., 47 (1951), 207-216. [Annotated scanned copy]
Henry 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. 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).
Michael Gilleland, Some Self-Similar Integer Sequences
E. Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 84.
M. D. Hirschhorn, Some formulas for partitions into squares, Discrete Math, 211 (2000), pp. 225-228. [From _Ant King_, Oct 05 2010]
Index entries for sequences related to sublattices
Index entries for sequences related to sums of squares
Index entries for "core" sequences

Cf. A002654, A001481, A025427, A025428, A063725, A025426, A000290.
Cf. A000925, A247367.
Equivalent sequences for other numbers of squares: A010052 (1), A000164 (3), A002635 (4), A000174 (5).

a000161 n =
   sum $ map (a010052 . (n -)) $ takeWhile (<= n `div` 2) a000290_list
a000161_list = map a000161 [0..]
-- Reinhard Zumkeller, Aug 16 2011

A000161 := proc(n) local i,j,ans; ans := 0; for i from 0 to n do for j from i to n do if i^2+j^2=n then ans := ans+1 fi od od; RETURN(ans); end; [ seq(A000161(i), i=0..50) ];
A000161 := n -> nops( numtheory[sum2sqr](n) ); # M. F. Hasler, Nov 23 2007

Length[PowersRepresentations[ #,2,2]] &/@Range[0,150] (* Ant King, Oct 05 2010 *)

a(n)=sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1))) \\ for illustrative purpose

A000161(n)=sum(k=sqrtint((n-1)\2)+1,sqrtint(n),issquare(n-k^2)) \\ Charles R Greathouse IV, Mar 21 2014, improves earlier code by M. F. Hasler, Nov 23 2007

A000161(n)=#sum2sqr(n) \\ See A133388 for sum2sqr(). - M. F. Hasler, May 13 2018

from math import prod
from sympy import factorint
def A000161(n):
    f = factorint(n)
    return int(not any(e&1 for e in f.values())) + (((m:=prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in f.items()))+((((~n & n-1).bit_length()&1)<<1)-1 if m&1 else 0))>>1) if n else 1 # Chai Wah Wu, Sep 08 2022

a(n) = card { { a,b } c N | a^2+b^2 = n }. - M. F. Hasler, Nov 23 2007

Let f(n)= the number of divisors of n that are congruent to 1 modulo 4 minus the number of its divisors that are congruent to 3 modulo 4, and define delta(n) to be 1 if n is a perfect square and 0 otherwise. Then a(n)=1/2 (f(n)+delta(n)+delta(1/2 n)). - Ant King, Oct 05 2010

A022544 Numbers that are not the sum of 2 squares.

Original entry on oeis.org

3, 6, 7, 11, 12, 14, 15, 19, 21, 22, 23, 24, 27, 28, 30, 31, 33, 35, 38, 39, 42, 43, 44, 46, 47, 48, 51, 54, 55, 56, 57, 59, 60, 62, 63, 66, 67, 69, 70, 71, 75, 76, 77, 78, 79, 83, 84, 86, 87, 88, 91, 92, 93, 94, 95, 96, 99, 102, 103, 105, 107, 108, 110, 111, 112, 114, 115, 118, 119, 120, 123, 124, 126, 127, 129, 131, 132, 133, 134, 135, 138, 139, 140, 141, 142, 143, 147, 150, 151, 152, 154, 155, 156, 158, 159, 161, 163, 165, 166, 167, 168, 171, 172, 174, 175, 176, 177, 179, 182, 183, 184, 186, 187, 188, 189, 190, 191, 192, 195, 198, 199

Offset: 1

N. J. A. Sloane

Conjecture: if k is not the sum of 2 squares then sigma(k) == 0 (mod 4) (the converse does not hold, as demonstrated by the entries in A025303). - Benoit Cloitre, May 19 2002
Numbers having some prime factor p == 3 (mod 4) to an odd power. sigma(n) == 0 (mod 4) because of this prime factor. Every k == 3 (mod 4) is a term. First differences are always 1, 2, 3 or 4, each occurring infinitely often. - David W. Wilson, Mar 09 2005
Complement of A000415 in the nonsquare positive integers A000037. - Max Alekseyev, Jan 21 2010
Integers with an equal number of 4k+1 and 4k+3 divisors. - Ant King, Oct 05 2010
A000161(a(n)) = 0; A070176(a(n)) > 0; A046712 is a subsequence. - Reinhard Zumkeller, Feb 04 2012, Aug 16 2011
There are arbitrarily long runs of consecutive terms. Record runs start at 3, 6, 21, 75, ... (A260157). - Ivan Neretin, Nov 09 2015
From Klaus Purath, Sep 04 2023: (Start)
There are no squares in this sequence.
There are also no numbers of the form n^2 + 1 (A002522) or n^2 + 4 (A087475).
Every term a(n) raised to an odd power belongs to the sequence just as every product of an odd number of terms. This is also true for all integer sequences represented by the indefinite binary quadratic forms a(n)*x^2 - y^2. These sequences also do not contain squares. (End)

Steven R. Finch, Mathematical Constants, Cambridge, 2003, pp. 98-104.

T. D. Noe, Table of n, a(n) for n = 1..10000
Steven R. Finch, Landau-Ramanujan Constant [Broken link]
Steven R. Finch, Landau-Ramanujan Constant [From the Wayback machine]
Index entries for sequences related to sums of squares

Complement of A001481; subsequence of A111909.

Cf. A018825, A025284, A000404, A007692.

import Data.List (elemIndices)
a022544 n = a022544_list !! (n-1)
a022544_list = elemIndices 0 a000161_list
-- Reinhard Zumkeller, Aug 16 2011

[n: n in [0..160] | NormEquation(1, n) eq false]; // Vincenzo Librandi, Jan 15 2017

Select[Range[199], Length[PowersRepresentations[ #, 2, 2]] == 0 &] (* Ant King, Oct 05 2010 *)
Select[Range[200],SquaresR[2,#]==0&] (* Harvey P. Dale, Apr 21 2012 *)

for(n=0,200,if(sum(i=0,n,sum(j=0,i,if(i^2+j^2-n,0,1)))==0,print1((n),",")))

is(n)=if(n%4==3, return(1)); my(f=factor(n)); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ Charles R Greathouse IV, Sep 01 2015

def aupto(lim):
  squares = [k*k for k in range(int(lim**.5)+2) if k*k <= lim]
  sum2sqs = set(a+b for i, a in enumerate(squares) for b in squares[i:])
  return sorted(set(range(lim+1)) - sum2sqs)
print(aupto(199)) # Michael S. Branicky, Mar 06 2021

from itertools import count, islice
from sympy import factorint
def A022544_gen(): # generator of terms
    return filter(lambda n:any(p & 3 == 3 and e & 1 for p, e in factorint(n).items()),count(0))
A022544_list = list(islice(A022544_gen(),30)) # Chai Wah Wu, Jun 28 2022

Limit_{n->oo} a(n)/n = 1.

More terms from Benoit Cloitre, May 19 2002

A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 27, 29, 30, 32, 33, 34, 35, 36, 37, 38, 40, 41, 42, 43, 44, 45, 46, 48, 49, 50, 51, 52, 53, 54, 56, 57, 58, 59, 61, 62, 64, 65, 66, 67, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 81, 82, 83

Offset: 1

N. J. A. Sloane

nonn

An equivalent definition: numbers of the form x^2 + y^2 + z^2 with x,y,z >= 0.
Bourgain studies "the spatial distribution of the representation of a large integer as a sum of three squares, on the small and critical scale as well as their electrostatic energy. The main results announced give strong evidence to the thesis that the solutions behave randomly. This is in sharp contrast to what happens with sums of two or four or more square." Sums of two nonzero squares are A000404. - Jonathan Vos Post, Apr 03 2012
The multiplicities for a(n) (if 0 <= x <= y <= z) are given as A000164(a(n)), n >= 1. Compare with A005875(a(n)) for integer x, y and z, and order taken into account. - Wolfdieter Lang, Apr 08 2013
a(n)^k is a member of this sequence for any k > 1. - Boris Putievskiy, May 05 2013
The selection rule for the planes with Miller indices (hkl) to undergo X-ray diffraction in a simple cubic lattice is h^2+k^2+l^2 = N where N is a term of this sequence. See A004014 for f.c.c. lattice. - Mohammed Yaseen, Nov 06 2022

			a(1) = 0 = 0^2 + 0^2 + 0^2. A005875(0) = 1 = A000164(0).
a(9) = 9 = 0^2 + 0^2 + 3^2 =  1^2 +  2^2 + 2^2. A000164(9) = 2. A000164(9) = 30 = 2*3 + 8*3 (counting signs and order). - _Wolfdieter Lang_, Apr 08 2013

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 107.
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 37.
R. K. Guy, Unsolved Problems in Number Theory, Springer, 1st edition, 1981. See section C20.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 311.

T. D. Noe, Table of n, a(n) for n = 1..10000
Jean Bourgain, Peter Sarnak and Zeév Rudnick, Local statistics of lattice points on the sphere, arXiv:1204.0134 [math.NT], 2012-2015.
J. W. Cogdell, On sums of three squares, Journal de théorie des nombres de Bordeaux, 15 no. 1 (2003), p. 33-44.
L. E. Dickson, Integers represented by positive ternary quadratic forms, Bull. Amer. Math. Soc. 33 (1927), 63-70. See Theorem I.
Eric T. Mortenson, A Kronecker-type identity and the representations of a number as a sum of three squares, arXiv:1702.01627 [math.NT], 2017.
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91. [?Broken link]
P. Pollack, Analytic and Combinatorial Number Theory Course Notes, p. 91.
Eric Weisstein's World of Mathematics, Square Number
Index entries for sequences related to sums of squares

Union of A000290, A000404 and A000408 (common elements).
Union of A000290, A000415 and A000419 (disjunct sets).
Complement of A004215.
Cf. A005875 (number of representations if x, y and z are integers).
Cf. A047449, A034043, A034044, A034045, A034046, A034047, A065883, A072400, A072401, A071374, A002828, A001481, A125084, A000164.

isA000378 := proc(n) # return true or false depending on n being in the list
    local x,y ;
    for x from 0 do
        if 3*x^2 > n then
            return false;
        end if;
        for y from x do
            if x^2+2*y^2 > n then
                break;
            else
                if issqr(n-x^2-y^2) then
                    return true;
                end if;
            end if;
        end do:
    end do:
end proc:
A000378 := proc(n) # generate A000378(n)
    option remember;
    local a;
    if n = 1 then
        0;
    else
        for a from procname(n-1)+1 do
            if isA000378(a) then
                return a;
            end if;
        end do:
    end if;
end proc:
seq(A000378(n),n=1..100) ; # R. J. Mathar, Sep 09 2015

okQ[n_] := If[EvenQ[k = IntegerExponent[n, 2]], m = n/2^k; Mod[m, 8] != 7, True]; Select[Range[0, 100], okQ] (* Jean-François Alcover, Feb 08 2016, adapted from PARI *)

isA000378(n)=my(k=valuation(n, 2)); if(k%2==0, n>>=k; n%8!=7, 1)

list(lim)=my(v=List(),k,t); for(x=0,sqrtint(lim\=1), for(y=0, min(sqrtint(lim-x^2),x), k=x^2+y^2; for(z=0,min(sqrtint(lim-k), y), listput(v,k+z^2)))); Set(v) \\ Charles R Greathouse IV, Sep 14 2015

def valuation(n, b):
    v = 0
    while n > 1 and n%b == 0: n //= b; v += 1
    return v
def ok(n): return n//4**valuation(n, 4)%8 != 7
print(list(filter(ok, range(84)))) # Michael S. Branicky, Jul 15 2021

from itertools import count, islice
def A000378_gen(): # generator of terms
    return filter(lambda n:n>>2*(bin(n)[:1:-1].index('1')//2) & 7 < 7, count(1))
A000378_list = list(islice(A000378_gen(),30)) # Chai Wah Wu, Jun 27 2022

def A000378(n):
    def f(x): return n-1+sum(((x>>(i<<1))-7>>3)+1 for i in range(x.bit_length()>>1))
    m, k = n-1, f(n-1)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 14 2025

Legendre: a nonnegative integer is a sum of three squares iff it is not of the form 4^k m with m == 7 (mod 8).
n^(2k+1) is in the sequence iff n is in the sequence. - Ray Chandler, Feb 03 2009
Complement of A004215; complement of A000302(i)*A004771(j), i,j>=0. - Boris Putievskiy, May 05 2013
a(n) = 6n/5 + O(log n). - Charles R Greathouse IV, Mar 14 2014

More terms from Ray Chandler, Sep 05 2004

A008784 Numbers k such that sqrt(-1) mod k exists; or, numbers that are primitively represented by x^2 + y^2.

Original entry on oeis.org

1, 2, 5, 10, 13, 17, 25, 26, 29, 34, 37, 41, 50, 53, 58, 61, 65, 73, 74, 82, 85, 89, 97, 101, 106, 109, 113, 122, 125, 130, 137, 145, 146, 149, 157, 169, 170, 173, 178, 181, 185, 193, 194, 197, 202, 205, 218, 221, 226, 229, 233, 241, 250, 257, 265, 269, 274, 277, 281, 289

Offset: 1

N. J. A. Sloane, Olivier Gérard

nonn

Numbers whose prime divisors are all congruent to 1 mod 4, with the exception of at most a single factor of 2. - Franklin T. Adams-Watters, Sep 07 2008
In appears that {a(n)} is the set of proper divisors of numbers of the form m^2+1. - Kaloyan Todorov (kaloyan.todorov(AT)gmail.com), Mar 25 2009 [This conjecture is correct. - Franklin T. Adams-Watters, Oct 07 2009]
If a(n) is a term of this sequence, then so too are all of its divisors (Euler). - Ant King, Oct 11 2010
From Richard R. Forberg, Mar 21 2016: (Start)
For a given a(n) > 2, there are 2^k solutions to sqrt(-1) mod n (for some k >= 1), and 2^(k-1) solutions primitively representing a(n) by x^2 + y^2.
Record setting values for the number of solutions (i.e., the next higher k values), occur at values for a(n) given by A006278.
A224450 and A224770 give a(n) values with exactly one and exactly two solutions, respectively, primitively representing integers as x^2 + y^2.
The 2^k different solutions for sqrt(-1) mod n can written as values for j, with j <= n, such that integers r = sqrt(n*j-1). However, the set of j values (listed from smallest to largest) transform into themselves symmetrically (i.e., largest to smallest) when the solutions are written as n-r. When the same 2^k solutions are written as r-j, it is clear that only 2^(k-1) distinct and independent solutions exist. (End)
Lucas uses the fact that there are no multiples of 3 in this sequence to prove that one cannot have an equilateral triangle on the points of a square lattice. - Michel Marcus, Apr 27 2020
For n > 1, terms are precisely the numbers such that there is at least one pair (m,k) where m + k = a(n), and m*k == 1 (mod a(n)), m > 0 and m <= k. - Torlach Rush, Oct 18 2020
A pair (s,t) such that s+t = a(n) and s*t == +1 (mod a(n)) as above is obtained from a square root of -1 (mod a(n)) for s and t = a(n)-s. - Joerg Arndt, Oct 24 2020
The Diophantine equation x^2 + y^2 = z^5 + z with gcd(x, y, z) = 1 has solutions iff z is a term of this sequence. See Gardiner reference, Olympiad links and A340129. - Bernard Schott, Jan 17 2021
Except for 1, numbers of the form a + b + 2*sqrt(a*b - 1) for positive integers a,b such that a*b-1 is a square. - Davide Rotondo, Nov 10 2024

B. C. Berndt & R. A. Rankin, Ramanujan: Letters and Commentary, see p. 176; AMS Providence RI 1995.
J. W. S. Cassels, Rational Quadratic Forms, Cambridge, 1978.
Leonard Eugene Dickson, History of the Theory Of Numbers, Volume II: Diophantine Analysis, Chelsea Publishing Company, 1992, pp.230-242.
A. Gardiner, The Mathematical Olympiad Handbook: An Introduction to Problem Solving, Oxford University Press, 1997, reprinted 2011, Problem 6 pp. 63 and 167-168 (1985).
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, Ch. 20.2-3.

T. D. Noe, Table of n, a(n) for n = 1..1000
J.-P. Allouche and F. M. Dekking, Generalized Beatty sequences and complementary triples, arXiv:1809.03424 [math.NT], 2018.
British Mathematical Olympiad, 1985 - Problem 6.
Édouard Lucas, Théorème sur la géométrie des quinconces, Nouvelles annales de mathématiques : journal des candidats aux écoles polytechnique et normale, Série 2, Tome 17 (1878), p. 129-130.
P. Cho-Ho Lam, Representation of integers using a^2+b^2-dc^2, J. Int. Seq. 18 (2015) 15.8.6, Theorems 2 and 3.
Richard J. Mathar, Construction of Bhaskara pairs, arXiv:1703.01677 [math.NT], 2017.
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references).
Index to sequences related to Olympiads.

Apart from the first term, a subsequence of A000404.

Cf. A001481, A022544, A020893, A037942, A034023, A057756, A076948, A045673, A004613, A340129, A192450 (complement).

import Data.List.Ordered (union)
a008784 n = a008784_list !! (n-1)
a008784_list = 1 : 2 : union a004613_list (map (* 2) a004613_list)
-- Reinhard Zumkeller, Oct 25 2015

with(numtheory); [seq(mroot(-1,2,p),p=1..300)];

data=Flatten[FindInstance[x^2+y^2==# && 0<=x<=# && 0<=y<=# && GCD[x,y]==1,{x,y},Integers]&/@Range[289],1]; x^2+y^2/.data//Union (* Ant King, Oct 11 2010 *)
Select[Range[289], And @@ (Mod[#, 4] == 1 & ) /@ (fi = FactorInteger[#]; If[fi[[1]] == {2, 1}, Rest[fi[[All, 1]]], fi[[All, 1]]])&] (* Jean-François Alcover, Jul 02 2012, after Franklin T. Adams-Watters *)

is(n)=if(n%2==0,if(n%4,n/=2,return(0)));n==1||vecmax(factor(n)[,1]%4)==1 \\ Charles R Greathouse IV, May 10 2012

list(lim)=my(v=List([1,2]),t); lim\=1; for(x=2,sqrtint(lim-1), t=x^2; for(y=0,min(x-1,sqrtint(lim-t)), if(gcd(x,y)==1, listput(v,t+y^2)))); Set(v) \\ Charles R Greathouse IV, Sep 06 2016

for(n=1,300,if(issquare(Mod(-1, n)),print1(n,", "))); \\ Joerg Arndt, Apr 27 2020

Checked by T. D. Noe, Apr 19 2007

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

A055025 Norms of Gaussian primes.

Original entry on oeis.org

2, 5, 9, 13, 17, 29, 37, 41, 49, 53, 61, 73, 89, 97, 101, 109, 113, 121, 137, 149, 157, 173, 181, 193, 197, 229, 233, 241, 257, 269, 277, 281, 293, 313, 317, 337, 349, 353, 361, 373, 389, 397, 401, 409, 421, 433, 449, 457, 461, 509, 521, 529, 541, 557, 569

Offset: 1

N. J. A. Sloane, Jun 09 2000

This is the range of the norm a^2 + b^2 of Gaussian primes a + b i. A239621 lists for each norm value a(n) one of the Gaussian primes as a, b with a >= b >= 0. In A239397, any of these (a, b) is followed by (b, a), except for a = b = 1. - Wolfdieter Lang, Mar 24 2014, edited by M. F. Hasler, Mar 09 2018
From Jean-Christophe Hervé, May 01 2013: (Start)
The present sequence is related to the square lattice, and to its division in square sublattices. 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. Then A001481 (norms of Gaussian integers) 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. The present sequence gives the "prime divisors" of the square lattice.
Similarly, A055664 (Norms of Eisenstein-Jacobi primes) is the sequence of "prime divisors" of the hexagonal lattice. (End)
The sequence is formed of 2, the prime numbers of form 4k+1, and the square of other primes (of form 4k+3). These are the primitive elements of A001481. With 0 and 1, they are the numbers that are uniquely decomposable in the sum of two squares. - Jean-Christophe Hervé, Nov 17 2013

			There are 8 Gaussian primes of norm 5, +-1+-2i and +-2+-i, but only two inequivalent ones (2+-i). In A239621 2+i is listed as 2, 1.

R. K. Guy, Unsolved Problems in Number Theory, A16.
L. W. Reid, The Elements of the Theory of Algebraic Numbers, MacMillan, NY, 1910, see Chap. V.

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
Eric Weisstein's World of Mathematics, Gaussian prime
Wikipedia, Table of Gaussian integer factorizations
Index entries for Gaussian integers and primes

Cf. A055026, A055027, A055028, A055029, A055664-A055666, A001481.

Cf. A239397, A239621 (Gaussian primes).

Union[(#*Conjugate[#] & )[ Select[Flatten[Table[a + b*I, {a, 0, 23}, {b, 0, 23}]], PrimeQ[#, GaussianIntegers -> True] & ]]][[1 ;; 55]] (* Jean-François Alcover, Apr 08 2011 *)
(* Or, from formula: *) maxNorm = 569; s1 = Select[Range[1, maxNorm, 4], PrimeQ]; s3 = Select[Range[3, Sqrt[maxNorm], 4], PrimeQ]^2; Union[{2}, s1, s3]  (* Jean-François Alcover, Dec 07 2012 *)

list(lim)=my(v=List()); if(lim>=2, listput(v,2)); forprime(p=3,sqrtint(lim\1), if(p%4==3, listput(v,p^2))); forprime(p=5,lim, if(p%4==1, listput(v,p))); Set(v) \\ Charles R Greathouse IV, Feb 06 2017

isA055025(n)=(isprime(n) && n%4<3) || (issquare(n, &n) && isprime(n) && n%4==3) \\ Jianing Song, Aug 15 2023, based on Charles R Greathouse IV's program for A055664

Consists of 2; rational primes = 1 (mod 4) [A002144]; and squares of rational primes = 3 (mod 4) [A002145^2].

a(n) ~ 2n log n. - Charles R Greathouse IV, Feb 06 2017

More terms from Larry Reeves (larryr(AT)acm.org), Oct 03 2000

A031363 Positive numbers of the form x^2 + xy - y^2; or, of the form 5x^2 - y^2.

Original entry on oeis.org

1, 4, 5, 9, 11, 16, 19, 20, 25, 29, 31, 36, 41, 44, 45, 49, 55, 59, 61, 64, 71, 76, 79, 80, 81, 89, 95, 99, 100, 101, 109, 116, 121, 124, 125, 131, 139, 144, 145, 149, 151, 155, 164, 169, 171, 176, 179, 180, 181, 191, 196, 199, 205, 209, 211, 220, 225, 229, 236

Offset: 1

N. J. A. Sloane

5x^2 - y^2 has discriminant 20, x^2 + xy - y^2 has discriminant 5. - N. J. A. Sloane, May 30 2014
Representable as x^2 + 3xy + y^2 with 0 <= x <= y. - Benoit Cloitre, Nov 16 2003
Numbers k such that x^2 - 3xy + y^2 + k = 0 has integer solutions. - Colin Barker, Feb 04 2014
Numbers k such that x^2 - 7xy + y^2 + 9k = 0 has integer solutions. - Colin Barker, Feb 10 2014
Also positive numbers of the form x^2 - 5y^2. - Jon E. Schoenfield, Jun 03 2022

M. Baake, "Solution of coincidence problem ...", in R. V. Moody, ed., Math. of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44.

Robert Israel and Vincenzo Librandi, Table of n, a(n) for n = 1..10000
M. Baake, Solution of the coincidence problem in dimensions d <= 4, arxiv:math/0605222 [math.MG], 2006.
M. Baake and R. V. Moody, Similarity submodules and semigroups in Quasicrystals and Discrete Geometry, ed. J. Patera, Fields Institute Monographs, vol. 10 AMS, Providence, RI (1998) pp. 1-13.
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).
Norbert Hungerbühler and Maciej Smela, Geometric approach to the Diophantine equation x^2 + x*y - y^2 = m, hal-04835410, 2024. See p. 18.
N. J. A. Sloane, My favorite integer sequences, in Sequences and their Applications (Proceedings of SETA '98).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)

Numbers representable as x^2 + k*x*y + y^2 with 0 <= x <= y, for k=0..9: A001481(k=0), A003136(k=1), A000290(k=2), this sequence, A084916(k=4), A243172(k=5), A242663(k=6), A243174(k=7), A243188(k=8), A316621(k=9).
See A035187 for number of representations.
Primes in this sequence: A038872, also A141158.
For a list of sequences giving numbers and/or primes represented by binary quadratic forms, see the "Binary Quadratic Forms and OEIS" link.
See also the related sequence A263849 based on a theorem of Maass.

select(t -> nops([isolve(5*x^2-y^2=t)])>0, [$1..1000]); # Robert Israel, Jun 12 2014

ok[n_] := Resolve[Exists[{x, y}, Element[x|y, Integers], n == 5*x^2-y^2]]; Select[Range[236], ok]
(* or, for a large number of terms: *)
max = 60755 (* max=60755 yields 10000 terms *); A031363 = {}; xm = 1;
While[T = A031363; A031363 = Table[5*x^2 - y^2, {x, 1, xm}, {y, 0, Floor[ x*Sqrt[5]]}] // Flatten // Union // Select[#, # <= max&]&; A031363 != T, xm = 2*xm]; A031363  (* Jean-François Alcover, Mar 21 2011, updated Mar 17 2018 *)

select(x -> x, direuler(p=2,101,1/(1-(kronecker(5,p)*(X-X^2))-X)), 1) \\ Fixed by Andrey Zabolotskiy, Jul 30 2020, after hints by Colin Barker, Jun 18 2014, and Michel Marcus

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

seq(M,k=3) = { \\ assume k >= 0
setintersect([1..M], setbinop((x,y)->x^2 + k*x*y + y^2, [0..1+sqrtint(M)]));
};
seq(236) \\ Gheorghe Coserea, Jul 29 2018

from itertools import count, islice
from sympy import factorint
def A031363_gen(): # generator of terms
    return filter(lambda n:all(not((1 < p % 5 < 4) and e & 1) for p, e in factorint(n).items()),count(1))
A031363_list = list(islice(A031363_gen(),30)) # Chai Wah Wu, Jun 28 2022

Consists exactly of numbers in which primes == 2 or 3 mod 5 occur with even exponents.

Indices of the nonzero terms in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = 5.

More terms from Erich Friedman

b-file corrected and extended by Robert Israel, Jun 12 2014

A051533 Numbers that are the sum of two positive triangular numbers.

Original entry on oeis.org

2, 4, 6, 7, 9, 11, 12, 13, 16, 18, 20, 21, 22, 24, 25, 27, 29, 30, 31, 34, 36, 37, 38, 39, 42, 43, 46, 48, 49, 51, 55, 56, 57, 58, 60, 61, 64, 65, 66, 67, 69, 70, 72, 73, 76, 79, 81, 83, 84, 87, 88, 90, 91, 92, 93, 94, 97, 99, 100, 101, 102, 106, 108

Offset: 1

Klaus Strassburger (strass(AT)ddfi.uni-duesseldorf.de)

Numbers n such that 8n+2 is in A085989. - Robert Israel, Mar 06 2017

			666 is in the sequence because we can write 666 = 435 + 231 = binomial(22,2) + binomial(30,2).

T. D. Noe, Table of n, a(n) for n = 1..1000
Eric Weisstein's World of Mathematics, Fermat's Polygonal Number Theorem

Cf. A000217, A020756 (sums of two triangular numbers), A001481 (sums of two squares), A007294, A051611 (complement).
Cf. A061336: minimal number of triangular numbers that sum up to n.
Cf. A085989.

a051533 n = a051533_list !! (n-1)
a051533_list = filter ((> 0) . a053603) [1..]
-- Reinhard Zumkeller, Jun 28 2013

isA051533 := proc(n)
    local a,ta;
    for a from 1 do
        ta := A000217(a) ;
        if 2*ta > n then
            return false;
        end if;
        if isA000217(n-ta) then
            return true;
        end if;
    end do:
end proc:
for n from 1 to 200 do
    if isA051533(n) then
        printf("%d,",n) ;
    end if;
end do: # R. J. Mathar, Dec 16 2015

f[k_] := If[!
   Head[Reduce[m (m + 1) + n (n + 1) == 2 k && 0 < m && 0 < n, {m, n},
       Integers]] === Symbol, k, 0]; DeleteCases[Table[f[k], {k, 1, 108}], 0] (* Ant King, Nov 22 2010 *)
nn=50; tri=Table[n(n+1)/2, {n,nn}]; Select[Union[Flatten[Table[tri[[i]]+tri[[j]], {i,nn}, {j,i,nn}]]], #<=tri[[-1]] &]
With[{nn=70},Take[Union[Total/@Tuples[Accumulate[Range[nn]],2]],nn]] (* Harvey P. Dale, Jul 16 2015 *)

is(n)=for(k=ceil((sqrt(4*n+1)-1)/2),(sqrt(8*n-7)-1)\2, if(ispolygonal(n-k*(k+1)/2, 3), return(1))); 0 \\ Charles R Greathouse IV, Jun 09 2015

A053603(a(n)) > 0. - Reinhard Zumkeller, Jun 28 2013

A061336(a(n)) = 2. - M. F. Hasler, Mar 06 2017

A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

Original entry on oeis.org

1, 2, 5, 4, 3, 10, 13, 8, 25, 6, 17, 20, 7, 26, 15, 16, 11, 50, 29, 12, 65, 34, 37, 40, 9, 14, 125, 52, 19, 30, 41, 32, 85, 22, 39, 100, 23, 58, 35, 24, 31, 130, 53, 68, 75, 74, 61, 80, 169, 18, 55, 28, 43, 250, 51, 104, 145, 38, 73, 60, 47, 82, 325, 64, 21, 170, 89, 44, 185, 78, 97, 200, 59, 46, 45, 116, 221, 70, 101

Offset: 1

Antti Karttunen, Feb 01 2016

Lexicographically earliest self-inverse permutation of natural numbers where each prime of the form 4k+1 is replaced by a prime of the form 4k+3 and vice versa, with the composite numbers determined by multiplicativity.
Fully multiplicative with a(p_n) = p_{A267100(n)} = A267101(n).
Maps each term of A004613 to some term of A004614, each (nonzero) term of A001481 to some term of A268377 and each term of A004431 to some term of A268378 and vice versa.
Sequences A072202 and A078613 are closed with respect to this permutation.

Antti Karttunen, Table of n, a(n) for n = 1..10000
Index entries for sequences that are permutations of the natural numbers

Cf. A000035, A000040, A000720, A010051, A020639, A032742, A267100, A267101, A354102 (Möbius transform), A354103 (inverse Möbius transform), A354192 (fixed points).
Cf. A002144, A002145, A005094, A065338, A072202, A078613, A079635.
Cf. A001481, A268377, A004431, A268378, A004613, A004614.
Cf. also A108548.

up_to = 2^16;
A267097list(up_to) = { my(v=vector(up_to),i=0,c=0); forprime(p=2,prime(up_to), if(1==(p%4), c++); i++; v[i] = c); (v); };
v267097 = A267097list(up_to);
A267097(n) = v267097[n];
A267098(n) = ((n-1)-A267097(n));
list_primes_of_the_form(up_to,m,k) = { my(v=vector(up_to),i=0); forprime(p=2,, if(k==(p%m), i++; v[i] = p; if(i==up_to,return(v)))); };
v002144 = list_primes_of_the_form(2*up_to,4,1);
A002144(n) = v002144[n];
v002145 = list_primes_of_the_form(2*up_to,4,3);
A002145(n) = v002145[n];
A267101(n) = if(1==n,2,if(1==(prime(n)%4),A002145(A267097(n)),A002144(A267098(n))));
A267099(n) = { my(f=factor(n)); for(k=1,#f~,f[k,1] = A267101(primepi(f[k,1]))); factorback(f); }; \\ Antti Karttunen, May 18 2022
(Scheme, with memoization-macro definec)
(definec (A267099 n) (cond ((<= n 1) n) ((= 1 (A010051 n)) (A267101 (A000720 n))) (else (* (A267099 (A020639 n)) (A267099 (A032742 n))))))

a(1) = 1; after which, if n is k-th prime [= A000040(k)], then a(n) = A267101(k), otherwise a(A020639(n)) * a(A032742(n)).
Other identities. For all n >= 1:
a(A000040(n)) = A267101(n).
a(2*n) = 2*a(n).
a(3*n) = 5*a(n).
a(5*n) = 3*a(n).
a(7*n) = 13*a(n).
a(11*n) = 17*a(n).
etc. See examples in A267101.
A000035(n) = A000035(a(n)). [Preserves the parity of n.]
A005094(a(n)) = -A005094(n).
A079635(a(n)) = -A079635(n).

Verbal description prefixed to the name by Antti Karttunen, May 19 2022

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.

Original entry on oeis.org

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

Offset: 1

Zhi-Wei Sun, Apr 25 2018

nonn

Conjecture: a(n) > 0 for all n > 1. In other words, any integer n > 1 can be written as the sum of two squares and two central binomial coefficients.
It has been verified that a(n) > 0 for all n = 2..10^10.
See also A303539 and A303541 for related information.
Jiao-Min Lin (a student at Nanjing University) has verified a(n) > 0 for all 1 < n <= 10^11. - Zhi-Wei Sun, Jul 30 2022

			a(2) = 1 since 2 = 0^2 + 0^2 + binomial(2*0,0) + binomial(2*0,0).
a(10) = 2 with 10 = 2^2 + 2^2 + binomial(2*0,0) + binomial(2*0,0) = 1^2 + 1^2 + binomial(2*1,1) + binomial(2*2,2).
a(2435) = 1 with 2435 = 32^2 + 33^2 + binomial(2*4,4) + binomial(2*5,5).

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, A000984, A001481, A303233, A303234, A303338, A303363, A303389, A303393, A303399, A303428, A303401, A303432, A303434, A303539, A303541, A303543, A303601, A303639.

N:= 100: # for a(1)..a(N)
A:= Vector(N):
for b from 0 to floor(sqrt(N)) do
  for a from 0 to min(b, floor(sqrt(N-b^2))) do
    t:= a^2+b^2;
    for d from 0 do
      s:= t + binomial(2*d,d);
      if s > N then break fi;
      for c from 0 to d do
        u:= s + binomial(2*c,c);
        if u > N then break fi;
        A[u]:= A[u]+1;
od od od od:
convert(A,list); # Robert Israel, May 30 2018

SQ[n_]:=SQ[n]=IntegerQ[Sqrt[n]];
c[n_]:=c[n]=Binomial[2n,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]

cp's OEIS Frontend

A000161 Number of partitions of n into 2 squares.

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A022544 Numbers that are not the sum of 2 squares.

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A000378 Sums of three squares: numbers of the form x^2 + y^2 + z^2.

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A008784 Numbers k such that sqrt(-1) mod k exists; or, numbers that are primitively represented by x^2 + y^2.

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A064533 Decimal expansion of Landau-Ramanujan constant.

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A267099 Fully multiplicative involution swapping the positions of 4k+1 and 4k+3 primes: a(1) = 1; a(prime(k)) = A267101(k), a(xy) = a(x)a(y) for x, y > 1.

A303540 Number of ways to write n as a^2 + b^2 + binomial(2c,c) + binomial(2d,d), where a,b,c,d are nonnegative integers with a <= b and c <= d.