A022544 -id:A022544 - cp's OEIS frontend

A002145 Primes of the form 4*k + 3.

3, 7, 11, 19, 23, 31, 43, 47, 59, 67, 71, 79, 83, 103, 107, 127, 131, 139, 151, 163, 167, 179, 191, 199, 211, 223, 227, 239, 251, 263, 271, 283, 307, 311, 331, 347, 359, 367, 379, 383, 419, 431, 439, 443, 463, 467, 479, 487, 491, 499, 503, 523, 547, 563, 571

Offset: 1

N. J. A. Sloane

Or, odd primes p such that -1 is not a square mod p, i.e., the Legendre symbol (-1/p) = -1. [LeVeque I, p. 66]. - N. J. A. Sloane, Jun 28 2008
Primes which are not the sum of two squares, see the comment in A022544. - Artur Jasinski, Nov 15 2006
Natural primes which are also Gaussian primes. (It is a common error to refer to this sequence as "the Gaussian primes".)
Inert rational primes in the field Q(sqrt(-1)). - N. J. A. Sloane, Dec 25 2017
Numbers n such that the product of coefficients of (2n)-th cyclotomic polynomial equals -1. - Benoit Cloitre, Oct 22 2002
For p and q both belonging to the sequence, exactly one of the congruences x^2 = p (mod q), x^2 = q (mod p) is solvable, according to Gauss reciprocity law. - Lekraj Beedassy, Jul 17 2003
Also primes p that divide L((p-1)/2) or L((p+1)/2), where L(n) = A000032(n), the Lucas numbers. Union of A122869 and A122870. - Alexander Adamchuk, Sep 16 2006
Also odd primes p that divide ((p-1)!! + 1) or ((p-2)!! + 1). - Alexander Adamchuk, Nov 30 2006
Also odd primes p that divide ((p-1)!! - 1) or ((p-2)!! - 1). - Alexander Adamchuk, Apr 18 2007
This sequence is a proper subset of the set of the absolute values of negative fundamental discriminants (A003657). - Paul Muljadi, Mar 29 2008
Bernard Frénicle de Bessy discovered that such primes cannot be the hypotenuse of a Pythagorean triangle in opposition to primes of the form 4*n+1 (see A002144). - after Paul Curtz, Sep 10 2008
A079261(a(n)) = 1; complement of A145395. - Reinhard Zumkeller, Oct 12 2008
Subsequence of A007970. - Reinhard Zumkeller, Jun 18 2011
A151763(a(n)) = -1.
Primes p such that p XOR 2 = p - 2. Brad Clardy, Oct 25 2011 (Misleading in the sense that this is a formula for the super-sequence A004767. - R. J. Mathar, Jul 28 2014)
It appears that each term of A004767 is the mean of two terms of this subsequence of primes therein; cf. A245203. - M. F. Hasler, Jul 13 2014
Numbers n > 2 such that ((n-2)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 24 2016
Odd numbers n > 1 such that ((n-1)!!)^2 == 1 (mod n). - Thomas Ordowski, Jul 25 2016
Primes p such that (p-2)!! == (p-3)!! (mod p). - Thomas Ordowski, Jul 28 2016
See Granville and Martin for a discussion of the relative numbers of primes of the form 4k+1 and 4k+3. - Editors, May 01 2017
Sometimes referred to as Blum primes for their connection to A016105 and the Blum Blum Shub generator. - Charles R Greathouse IV, Jun 14 2018
Conjecture: a(n) for n > 4 can be written as a sum of 3 primes of the form 4k+1, which would imply that primes of the form 4k+3 >= 23 can be decomposed into a sum of 6 nonzero squares. - Thomas Scheuerle, Feb 09 2023

M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards Applied Math. Series 55, 1964 (and various reprintings), p. 870.
John H. Conway and Richard K. Guy, The Book of Numbers, New York: Springer-Verlag, 1996. See pp. 146-147.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, 5th ed., Oxford Univ. Press, 1979, p. 219, th. 252.
W. J. LeVeque, Topics in Number Theory. Addison-Wesley, Reading, MA, 2 vols., 1956, Vol. 1, p. 66.
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).
David Wells, The Penguin Dictionary of Curious and Interesting Numbers. Penguin Books, NY, 1986, Revised edition 1987. See p. 90.

Zak Seidov, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
M. Abramowitz and I. A. Stegun, eds., Handbook of Mathematical Functions, National Bureau of Standards, Applied Math. Series 55, Tenth Printing, 1972 [alternative scanned copy].
Dario Alpern, Gaussian primes.
Lenore Blum, Manuel Blum, and Mike Shub, A simple unpredictable pseudo-random number generator, SIAM Journal on Computing 15:2 (1 May 1986), pp. 364-383.
Andrew Granville and Greg Martin, Prime number races, arXiv:math/0408319 [math.NT], 2004.
Ernest G. Hibbs, Component Interactions of the Prime Numbers, Ph. D. Thesis, Capitol Technology Univ. (2022), see p. 33.
Lucas Lacasa, Bartolome Luque, Ignacio Gómez, and Octavio Miramontes, On a Dynamical Approach to Some Prime Number Sequences, Entropy 20.2 (2018): 131, also arXiv:1802.08349 [math.NT], 2018.
E. T. Ordman, Tables of the class number for negative prime discriminants, Deposited in Unpublished Mathematical Table file of Math. Comp. [Annotated scanned partial copy with notes]
Mario Raso, Integer Sequences in Cryptography: A New Generalized Family and its Application, Ph. D. Thesis, Sapienza University of Rome (Italy 2025). See p. 67.
Harry J. Smith, Gaussian Primes.
Ian Stewart, The Great Mathematical Problems, 2013.
Eric Weisstein's World of Mathematics, Gaussian Prime.
Eric Weisstein's World of Mathematics, Gaussian Integer.
Wolfram Research, The Gauss Reciprocity Law.
Index entries for Gaussian integers and primes.
Index to sequences related to decomposition of primes in quadratic fields.

Cf. A000032, A002144, A003657, A085992, A122869, A122870, A334912.
Cf. A000408, A005098, A095278, A016754.
Apart from initial term, same as A045326.
Cf. A016105.
Cf. A004614 (multiplicative closure).

a002145 n = a002145_list !! (n-1)
a002145_list = filter ((== 1) . a010051) [3, 7 ..]
-- Reinhard Zumkeller, Aug 02 2015, Sep 23 2011

[4*n+3 : n in [0..142] | IsPrime(4*n+3)]; // Arkadiusz Wesolowski, Nov 15 2013

A002145 := proc(n)
    option remember;
    if n = 1 then
        3;
    else
        a := nextprime(procname(n-1)) ;
        while a mod 4 <>  3 do
            a := nextprime(a) ;
        end do;
        return a;
    end if;
end proc:
seq(A002145(n),n=1..20) ; # R. J. Mathar, Dec 08 2011

Select[4Range[150] - 1, PrimeQ] (* Alonso del Arte, Dec 19 2013 *)
Select[ Prime@ Range[2, 110], Length@ PowersRepresentations[#^2, 2, 2] == 1 &] (* or *)
Select[ Prime@ Range[2, 110], JacobiSymbol[-1, #] == -1 &] (* Robert G. Wilson v, May 11 2014 *)

forprime(p=2,1e3,if(p%4==3,print1(p", "))) \\ Charles R Greathouse IV, Jun 10 2011

def A002145_list(n): return [p for p in prime_range(1, n + 1) if p % 4 == 3]  # Peter Luschny, Jul 29 2014

Remove from A000040 terms that are in A002313.
Intersection of A000040 and A004767. - Alonso del Arte, Apr 22 2014
From Vaclav Kotesovec, Apr 30 2020: (Start)
Product_{k>=1} (1 - 1/a(k)^2) = A243379.
Product_{k>=1} (1 + 1/a(k)^2) = A243381.
Product_{k>=1} (1 - 1/a(k)^3) = A334427.
Product_{k>=1} (1 + 1/a(k)^3) = A334426.
Product_{k>=1} (1 - 1/a(k)^4) = A334448.
Product_{k>=1} (1 + 1/a(k)^4) = A334447.
Product_{k>=1} (1 - 1/a(k)^5) = A334452.
Product_{k>=1} (1 + 1/a(k)^5) = A334451. (End)
From Vaclav Kotesovec, May 05 2020: (Start)
Product_{k>=1} (1 + 1/a(k)) / (1 + 1/A002144(k)) = Pi/(4*A064533^2) = 1.3447728438248695625516649942427635670667319092323632111110962...
Product_{k>=1} (1 - 1/a(k)) / (1 - 1/A002144(k)) = Pi/(8*A064533^2) = 0.6723864219124347812758324971213817835333659546161816055555481... (End)
Sum_{k >= 1} 1/a(k)^s = (1/2) * Sum_{n >= 1 odd numbers} moebius(n) * log(2 * (2^(n*s) - 1) * (n*s - 1)! * zeta(n*s) / (Pi^(n*s) * abs(EulerE(n*s - 1))))/n, s >= 3 odd number. - Dimitris Valianatos, May 20 2020

More terms from James Sellers, Apr 21 2000

A001481 Numbers that are the sum of 2 squares.

Original entry on oeis.org

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

A004767 a(n) = 4*n + 3.

Original entry on oeis.org

3, 7, 11, 15, 19, 23, 27, 31, 35, 39, 43, 47, 51, 55, 59, 63, 67, 71, 75, 79, 83, 87, 91, 95, 99, 103, 107, 111, 115, 119, 123, 127, 131, 135, 139, 143, 147, 151, 155, 159, 163, 167, 171, 175, 179, 183, 187, 191, 195, 199, 203, 207, 211, 215, 219, 223

Offset: 0

N. J. A. Sloane

Apart from initial term(s), dimension of the space of weight 2n cusp forms for Gamma_0(12).
Binary expansion ends 11.
These the numbers for which zeta(2*x+1) needs just 2 terms to be evaluated. - Jorge Coveiro, Dec 16 2004 [This comment needs clarification]
a(n) is the smallest k such that for every r from 0 to 2n - 1 there exist j and i, k >= j > i > 2n - 1, such that j - i == r (mod (2n - 1)), with (k, (2n - 1)) = (j,(2n - 1)) = (i, (2n - 1)) = 1. - Amarnath Murthy, Sep 24 2003
Complement of A004773. - Reinhard Zumkeller, Aug 29 2005
Any (4n+3)-dimensional manifold endowed with a mixed 3-Sasakian structure is an Einstein space with Einstein constant lambda = 4n + 2 [Theorem 3, p. 10 of Ianus et al.]. - Jonathan Vos Post, Nov 24 2008
Solutions to the equation x^(2*x) = 3*x (mod 4*x). - Farideh Firoozbakht, May 02 2010
Subsequence of A022544. - Vincenzo Librandi, Nov 20 2010
First differences of A084849. - Reinhard Zumkeller, Apr 02 2011
Numbers n such that {1, 2, 3, ..., n} is a losing position in the game of Nim. - Franklin T. Adams-Watters, Jul 16 2011
Numbers n such that there are no primes p that satisfy the relationship p XOR n = p + n. - Brad Clardy, Jul 22 2012
The XOR of all numbers from 1 to a(n) is 0. - David W. Wilson, Apr 21 2013
A089911(4*a(n)) = 4. - Reinhard Zumkeller, Jul 05 2013
First differences of A014105. - Ivan N. Ianakiev, Sep 21 2013
All triangular numbers in the sequence are congruent to {3, 7} mod 8. - Ivan N. Ianakiev, Nov 12 2013
Apart from the initial term, length of minimal path on an n-dimensional cubic lattice (n > 1) of side length 2, until a self-avoiding walk gets stuck. Construct a path connecting all 2n points orthogonally adjacent from the center, ending at the center. Starting at any point adjacent to the center, there are 2 steps to reach each of the remaining 2n - 1 points, resulting in path length 4n - 2 with a final step connecting the center, for a total path length of 4n - 1, comprising 4n points. - Matthew Lehman, Dec 10 2013
a(n-1), n >= 1, appears as first column in the triangles A238476 and A239126 related to the Collatz problem. - Wolfdieter Lang, Mar 14 2014
For the Collatz Conjecture, we identify two types of odd numbers. This sequence contains all the ascenders: where (3*a(n) + 1) / 2 is odd and greater than a(n). See A016813 for the descenders. - Jaroslav Krizek, Jul 29 2016

			G.f. = 3 + 7*x + 11*x^2 + 15*x^3 + 19*x^4 + 23*x^5 + 27*x^6 + 31*x^7 + ...

Alfred S. Posamentier, Math Charmers, Tantalizing Tidbits for the Mind, Prometheus Books, NY, 2003, page 85.
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999. See Theorem 8.1 on page 240.

Ivan Panchenko, Table of n, a(n) for n = 0..200
Guo-Niu Han, Enumeration of Standard Puzzles
Guo-Niu Han, Enumeration of Standard Puzzles [Cached copy]
Stere Ianus, Mihai Visinescu and Gabriel-Eduard Vilcu, Hidden symmetries and Killing tensors on curved spaces, arXiv:0811.3478 [math-ph], 2008. - _Jonathan Vos Post_, Nov 24 2008
Tanya Khovanova, Recursive Sequences
Juan F. Pulido, José L. Ramírez, and Andrés R. Vindas-Meléndez, Generating Trees and Fibonacci Polyominoes, arXiv:2411.17812 [math.CO], 2024. See p. 8.
William A. Stein, Dimensions of the spaces S_k(Gamma_0(N))
William A. Stein, The modular forms database
Index entries for linear recurrences with constant coefficients, signature (2,-1).

Cf. A004773, A005408, A008545 (partial products), A008586, A014105, A016813, A016825, A017137, A017629, A022544, A084849, A181049, A238476, A239126.
Cf. A017101 and A004771 (bisection: 3 and 7 mod 8).
Cf. A016838 (square).

a004767 = (+ 3) . (* 4)
a004767_list = [3, 7 ..]  -- Reinhard Zumkeller, Oct 03 2012

[4*n+3: n in [0..50]]; // Wesley Ivan Hurt, Jul 09 2014

Maple
```
seq( 3+4*n, n=0..100 );
```

4 Range[50] - 1 (* Wesley Ivan Hurt, Jul 09 2014 *)

a(n)=4*n+3 \\ Charles R Greathouse IV, Jul 28 2015

Vec((3+x)/(1-x)^2 + O(x^200)) \\ Altug Alkan, Jan 15 2016

for n in range(0,50): print(4*n+3, end=', ') # Stefano Spezia, Dec 12 2018

[4*n+3 for n in range(50)] # G. C. Greubel, Dec 09 2018

(0 to 59).map(4 *  + 3) // _Alonso del Arte, Dec 12 2018

G.f.: (3+x)/(1-x)^2. - Paul Barry, Feb 27 2003
a(n) = 2*a(n-1) - a(n-2) for n > 1, a(0) = 3, a(1) = 7. - Philippe Deléham, Nov 03 2008
a(n) = A017137(n)/2. - Reinhard Zumkeller, Jul 13 2010
a(n) = 8*n - a(n-1) + 2 for n > 0, a(0) = 3. - Vincenzo Librandi, Nov 20 2010
a(n) = A005408(A005408(n)). - Reinhard Zumkeller, Jun 27 2011
a(n) = 3 + A008586(n). - Omar E. Pol, Jul 27 2012
a(n) = A014105(n+1) - A014105(n). - Michel Marcus, Sep 21 2013
a(n) = A016813(n) + 2. - Jean-Bernard François, Sep 27 2013
a(n) = 4*n - 1, with offset 1. - Wesley Ivan Hurt, Mar 12 2014
From Ilya Gutkovskiy, Jul 29 2016: (Start)
E.g.f.: (3 + 4*x)*exp(x).
Sum_{n >= 0} (-1)^n/a(n) = (Pi + 2*log(sqrt(2) - 1))/(4*sqrt(2)) = A181049. (End)

A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

Original entry on oeis.org

1, 4, 4, 0, 4, 8, 0, 0, 4, 4, 8, 0, 0, 8, 0, 0, 4, 8, 4, 0, 8, 0, 0, 0, 0, 12, 8, 0, 0, 8, 0, 0, 4, 0, 8, 0, 4, 8, 0, 0, 8, 8, 0, 0, 0, 8, 0, 0, 0, 4, 12, 0, 8, 8, 0, 0, 0, 0, 8, 0, 0, 8, 0, 0, 4, 16, 0, 0, 8, 0, 0, 0, 4, 8, 8, 0, 0, 0, 0, 0, 8, 4, 8, 0, 0, 16, 0, 0, 0, 8, 8, 0, 0, 0, 0, 0, 0, 8, 4, 0, 12, 8

Offset: 0

N. J. A. Sloane

Number of points in square lattice on the circle of radius sqrt(n). Equivalently, number of Gaussian integers of norm n (cf. Conway-Sloane, p. 106).
Let b(n)=A004403(n), then Sum_{k=1..n} a(k)*b(n-k) = 1. - John W. Layman
Theta series of D_2 lattice.
Number 6 of the 74 eta-quotients listed in Table I of Martin (1996).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
The zeros in this sequence correspond to those integers with an equal number of 4k+1 and 4k+3 divisors, or equivalently to those that have at least one 4k+3 prime factor with an odd exponent (A022544). - Ant King, Mar 12 2013
If A(q) = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + ... denotes the o.g.f. of this sequence then the function F(q) := 1/4*(A(q^2) - A(q^4)) = ( Sum_{n >= 0} q^(2*n+1)^2 )^2 is the o.g.f. for counting the ways a positive integer n can be written as the sum of two positive odd squares. - Peter Bala, Dec 13 2013
Expansion coefficients of (2/Pi)*K, with the real quarter period K of elliptic functions, as series of the Jacobi nome q, due to (2/Pi)*K = theta_3(0,q)^2. See, e.g., Whittaker-Watson, p. 486. - Wolfdieter Lang, Jul 15 2016
Sum_{k=0..n} a(n) = A057655(n). Robert G. Wilson v, Dec 22 2016
Limit_{n->oo} (a(n)/n - Pi*log(n)) = A062089: Sierpinski's constant. - Robert G. Wilson v, Dec 22 2016
The mean value of a(n) is Pi, see A057655 for more details. - M. F. Hasler, Mar 20 2017

			G.f. = 1 + 4*q + 4*q^2 + 4*q^4 + 8*q^5 + 4*q^8 + 4*q^9 + 8*q^10 + 8*q^13 + 4*q^16 + 8*q^17 + 4*q^18 + 8*q^20 + 12*q^25 + 8*q^26 + ... . - _John Cannon_, Dec 30 2006

L. Comtet, Advanced Combinatorics, Reidel, 1974, p. 162, #16 (7), r(n).
J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 106.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.23).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15, p. 32, Lemma 2 (with the proof), p. 116, (9.10) first formula.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Cambridge, University Press, 1940, p. 133.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, p. 240, r(n).
W. König and J. Sprekels, Karl Weierstraß (1815-1897), Springer Spektrum, Wiesbaden, 2016, p. 186-187 and p. 280-281.
C. D. Olds, A. Lax and G. P. Davidoff, The Geometry of Numbers, Math. Assoc. Amer., 2000, p. 51.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, pages 244-245.
E. T. Whittaker and G. N. Watson, A Course of Modern Analysis, fourth edition, reprinted, 1958, Cambridge at the University Press.

T. D. Noe, Table of n, a(n) for n = 0..10000
G. E. Andrews, R. Lewis and Z.-G. Liu, An identity relating a theta series to a sum of Lambert series, Bull. London Math. Soc., 33 (2001), 25-31.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
S. Cooper and M. Hirschhorn, A combinatorial proof of a result from number theory, Integers 4 (2004), #A09.
Michael Gilleland, Some Self-Similar Integer Sequences
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math. 38 (1907), 1-62. (This sequence is called rho, see page 6)
M. D. Hirschhorn, Jacobi's Two-Square Theorem and Related Identities
M. D. Hirschhorn, Arithmetic Consequences of Jacobi's Two-Squares Theorem
M. D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Jacobi-Legendre letters, Correspondance mathématique entre Legendre et Jacobi, J. Reine Angew. Math. 80 (1875) 205-279, letter of September 9, 1828, p. 240-243, formula for 2K/Pi on p. 242.
Christian Kassel and Christophe Reutenauer, The zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1505.07229v3 [math.AG], 2015. [Note that a later version of this paper has a different title and different contents, and the number-theoretical part of the paper was moved to the publication which is next in this list.]
Christian Kassel and Christophe Reutenauer, Complete determination of the zeta function of the Hilbert scheme of n points on a two-dimensional torus, arXiv:1610.07793 [math.NT], 2016.
Christian Kassel, Christophe Reutenauer, The Fourier expansion of eta(z)eta(2z)eta(3z)/eta(6z), arXiv:1603.06357 [math.NT], 2016.
M. Kontsevich and D. Zagier, Periods, Institut des Hautes Etudes Scientifiques 2001 IHES/M/01/22. Published in B. Engquist and W. Schmid, editors, Mathematics Unlimited - 2001 and Beyond, 2 vols., Springer-Verlag, 2001, pp. 771-808, section 2.3. Example 3.
Y. Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
G. Nebe and N. J. A. Sloane, Home page for this lattice
F. Richman, Counting Gaussian integers in a disk
Grant Sanderson, Pi hiding in prime regularities, 3Blue1Brown video (2017).
N. J. A. Sloane et al., Binary Quadratic Forms and OEIS (Index to related sequences, programs, references)
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
G. Villemin, Sommes de 2 carrés
Eric Weisstein's World of Mathematics, Barnes-Wall Lattice
Eric Weisstein's World of Mathematics, Moebius Transform
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Eric Weisstein's World of Mathematics, Sum of Squares Function
Eric Weisstein's World of Mathematics, Theta Series
Zach Wissner-Gross a.k.a. The Riddler, Riddler Express: Can You Climb Your Way To Victory?, FiveThirtyEight, Sep 24 2021.
G. Xiao, Two squares
Index entries for sequences related to sums of squares
Index entries for sequences mentioned by Glaisher
Index entries for "core" sequences

Row d=2 of A122141 and of A319574, 2nd column of A286815.
Partial sums - 1 give A014198.
A071385 gives records; A071383 gives where records occur.
Cf. A001481, A004020, A005883, A057655 (partial sums), A057961, A057962, A002654,
A104271, A105673.

# JacobiTheta3 is defined in A000122.
A004018List(len) = JacobiTheta3(len, 2)
A004018List(102) |> println # Peter Luschny, Mar 12 2018

Basis( ModularForms( Gamma1(4), 1), 100) [1]; /* Michael Somos, Jun 10 2014 */

(sum(x^(m^2),m=-10..10))^2;
# Alternative:
A004018list := proc(len) series(JacobiTheta3(0, x)^2, x, len+1);
seq(coeff(%, x, j), j=0..len-1) end:
t1 := A004018list(102);
r2 := n -> t1[n+1]; # Peter Luschny, Oct 02 2018

SquaresR[2,Range[0,110]] (* Harvey P. Dale, Oct 10 2011 *)
a[ n_] := SquaresR[ 2, n]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q]^2, {q, 0, n}]; (* Michael Somos, Nov 15 2011 *)
a[ n_] := With[{m = InverseEllipticNomeQ @ q}, SeriesCoefficient[ EllipticK[ m] / (Pi/2), {q, 0, n}]]; (* Michael Somos, Jun 10 2014 *)
a[ n_] := If[ n < 1, Boole[n == 0], 4 Sum[ KroneckerSymbol[-4, d], {d, Divisors@n}]]; (* or *) a[ n_] := SeriesCoefficient[ QPochhammer[ q^2]^10/(QPochhammer[ q] QPochhammer[ q^4])^4, {q, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = polcoeff( 1 + 4 * sum( k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}; /* Michael Somos, Mar 14 2003 */

{a(n) = if( n<1, n==0, 4 * sumdiv( n, d, (d%4==1) - (d%4==3)))}; /* Michael Somos, Jul 19 2004 */

{a(n) = if( n<1, n==0, 2 * qfrep([ 1, 0; 0, 1], n)[n])}; /* Michael Somos, May 13 2005 */

a(n)=if(n==0,return(1)); my(f=factor(n)); 4*prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, if(f[i,2]%2 && f[i,1]>2, 0, 1))) \\ Charles R Greathouse IV, Sep 02 2015

from sympy import factorint
def a(n):
    if n == 0: return 1
    an = 4
    for pi, ei in factorint(n).items():
       if pi%4 == 1: an *= ei+1
       elif pi%4 == 3 and ei%2: return 0
    return an
print([a(n) for n in range(102)]) # Michael S. Branicky, Sep 24 2021

from math import prod
from sympy import factorint
def A004018(n): return prod(1 if p==2 else (e+1 if p&3==1 else (e+1)&1) for p, e in factorint(n).items())<<2 if n else 1 # Chai Wah Wu, Jul 07 2022, corrected Jun 21 2024.

Q = DiagonalQuadraticForm(ZZ, [1]*2)
Q.representation_number_list(102) # Peter Luschny, Jun 20 2014

Expansion of theta_3(q)^2 = (Sum_{n=-oo..+oo} q^(n^2))^2 = Product_{m>=1} (1-q^(2*m))^2 * (1+q^(2*m-1))^4; convolution square of A000122.
Factor n as n = p1^a1 * p2^a2 * ... * q1^b1 * q2^b2 * ... * 2^c, where the p's are primes == 1 (mod 4) and the q's are primes == 3 (mod 4). Then a(n) = 0 if any b is odd, otherwise a(n) = 4*(1 + a1)*(1 + a2)*...
G.f. = s(2)^10/(s(1)^4*s(4)^4), where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]
a(n) = 4*A002654(n), n > 0.
Expansion of eta(q^2)^10 / (eta(q) * eta(q^4))^4 in powers of q. - Michael Somos, Jul 19 2004
Expansion of ( phi(q)^2 + phi(-q)^2 ) / 2 in powers of q^2 where phi() is a Ramanujan theta function.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * 4 * w. - Michael Somos, Jul 19 2004
Euler transform of period 4 sequence [4, -6, 4, -2, ...]. - Michael Somos, Jul 19 2004
Moebius transform is period 4 sequence [4, 0, -4, 0, ...]. - Michael Somos, Sep 17 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i) f(t) where q = exp(2 Pi i t).
The constant sqrt(Pi)/Gamma(3/4)^2 produces the first 324 terms of the sequence when expanded in base exp(Pi), 450 digits of the constant are necessary. - Simon Plouffe, Mar 03 2011
a(n) = A004531(4*n). a(n) = 2*A105673(n), if n>0.
Let s = 16*q*(E1*E4^2/E2^3)^8 where Ek = Product_{n>=1} (1-q^(k*n)) (s=k^2 where k is elliptic k), then the g.f. is hypergeom([+1/2, +1/2], [+1], s) (expansion of 2/Pi*ellipticK(k) in powers of q). - Joerg Arndt, Aug 15 2011
Dirichlet g.f. Sum_{n>=1} a(n)/n^s = 4*zeta(s)*L_(-4)(s), where L is the D.g.f. of the (shifted) A056594. [Raman. J. 7 (2003) 95-127]. - R. J. Mathar, Jul 02 2012
a(n) = floor(1/(n+1)) + 4*floor(cos(Pi*sqrt(n))^2) - 4*floor(cos(Pi*sqrt(n/2))^2) + 8*Sum_{i=1..floor(n/2)} floor(cos(Pi*sqrt(i))^2)*floor(cos(Pi*sqrt(n-i))^2). - Wesley Ivan Hurt, Jan 09 2013
From Wolfdieter Lang, Aug 01 2016: (Start)
A Jacobi identity: theta_3(0, q)^2 = 1 + 4*Sum_{r>=0} (-1)^r*q^(2*r+1)/(1 - q^(2*r+1)). See, e.g., the Grosswald reference (p. 15, p. 116, but p. 32, Lemma 2 with the proof, has the typo r >= 1 instead of r >= 0 in the sum, also in the proof). See the link with the Jacobi-Legendre letter.
Identity used by Weierstraß (see the König-Sprekels book, p. 187, eq. (5.12) and p. 281, with references, but there F(x) from (5.11) on p. 186 should start with nu =1 not 0): theta_3(0, q)^2 = 1 + 4*Sum_{n>=1} q^n/(1 + q^(2*n)). Proof: similar to the one of the preceding Jacobi identity. (End)
a(n) = (4/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017
G.f.: Theta_3(q)^2 = hypergeometric([1/2, 1/2],[1],lambda(q)), with lambda(q) = Sum_{j>=1} A115977(j)*q^j. See the Kontsevich and Zagier link, with Theta -> Theta_3, z -> 2*z and q -> q^2. - Wolfdieter Lang, May 27 2018

A002654 Number of ways of writing n as a sum of at most two nonzero squares, where order matters; also (number of divisors of n of form 4m+1) - (number of divisors of form 4m+3).

Original entry on oeis.org

1, 1, 0, 1, 2, 0, 0, 1, 1, 2, 0, 0, 2, 0, 0, 1, 2, 1, 0, 2, 0, 0, 0, 0, 3, 2, 0, 0, 2, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 1, 3, 0, 2, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 1, 4, 0, 0, 2, 0, 0, 0, 1, 2, 2, 0, 0, 0, 0, 0, 2, 1, 2, 0, 0, 4, 0, 0, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 1, 0, 3, 2, 0, 0, 2, 0

Offset: 1

N. J. A. Sloane

Glaisher calls this E(n) or E_0(n). - N. J. A. Sloane, Nov 24 2018
Number of sublattices of Z X Z of index n that are similar to Z X Z; number of (principal) ideals of Z[i] of norm n.
a(n) is also one fourth of the number of integer solutions of n = x^2 + y^2 (order and signs matter, and 0 (without signs) is allowed). a(n) = N(n)/4, with N(n) from p. 147 of the Niven-Zuckermann reference. See also Theorem 5.12, p. 150, which defines a (strongly) multiplicative function h(n) which coincides with A056594(n-1), n >= 1, and N(n)/4 = sum(h(d), d divides n). - Wolfdieter Lang, Apr 19 2013
a(2+8*N) = A008441(N) gives the number of ways of writing N as the sum of 2 (nonnegative) triangular numbers for N >= 0. - Wolfdieter Lang, Jan 12 2017
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -4. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			4 = 2^2, so a(4) = 1; 5 = 1^2 + 2^2 = 2^2 + 1^2, so a(5) = 2.
x + x^2 + x^4 + 2*x^5 + x^8 + x^9 + 2*x^10 + 2*x^13 + x^16 + 2*x^17 + x^18 + ...
2 = (+1)^2 + (+1)^2 = (+1)^2 + (-1)^2  = (-1)^2 + (+1)^2 = (-1)^2 + (-1)^2. Hence there are 4 integer solutions, called N(2) in the Niven-Zuckerman reference, and a(2) = N(2)/4 = 1.  4 = 0^1 + (+2)^2 = (+2)^2 + 0^2 = 0^2 + (-2)^2 = (-2)^2 + 0^2. Hence N(4) = 4 and a(4) = N(4)/4 = 1. N(5) = 8, a(5) = 2. - _Wolfdieter Lang_, Apr 19 2013

J. M. Borwein, D. H. Bailey and R. Girgensohn, Experimentation in Mathematics, A K Peters, Ltd., Natick, MA, 2004. x+357 pp. See p. 194.
George Chrystal, Algebra: An elementary text-book for the higher classes of secondary schools and for colleges, 6th ed., Chelsea Publishing Co., New York, 1959, Part II, p. 346 Exercise XXI(17). MR0121327 (22 #12066)
Emil Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 15.
Ivan Niven and Herbert S. Zuckerman, An Introduction to the Theory of Numbers, New York: John Wiley, 1980, pp. 147 and 150.
Günter Scheja and Uwe Storch, Lehrbuch der Algebra, Tuebner, 1988, p. 251.
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).
James J. Tattersall, Elementary Number Theory in Nine Chapters, Cambridge University Press, 1999, page 89.
J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 340.

T. D. Noe, Table of n, a(n) for n = 1..10000
Michael Baake, Solution of the coincidence problem in dimensions d <= 4, in R. V. Moody, ed., The Mathematics of Long-Range Aperiodic Order, Kluwer 1997, pp. 9-44; arXiv:math/0605222 [math.MG], 2006.
Michael Baake and Uwe Grimm, Quasicrystalline combinatorics, 2002.
Shai Covo, Problem 3586, Crux Mathematicorum, Vol. 36, No. 7 (2010), pp. 461 and 463; Solution to Problem 3586 by the proposer, ibid., Vol. 37, No. 7 (2011), pp. 477-479.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167.
J. W. L. Glaisher, On the function chi(n), Quarterly Journal of Pure and Applied Mathematics, Vol. 20 (1884), pp. 97-167. [Annotated scanned copy]
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., Vol. 15 (1884), pp. 104-122.
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., Vol. 15 (1884), pp. 104-122. [Annotated scanned copy of pages 104-107 only]
J. W. L. Glaisher, On the representations of a number as the sum of two, four, six, eight, ten, and twelve squares, Quart. J. Math., Vol. 38 (1907), pp. 1-62 (see p. 4 and p. 8).
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.
Vaclav Kotesovec, Graph - the asymptotic ratio of Sum_{k=1..n} a(k)^2
Stephen C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., Vol. 6 (2002), pp. 7-149.
PeakMath, L-functions and the Langlands program (RH Saga S1E2), YouTube video (2024).
Srinivasa Ramanujan, Some formulas in the analytic theory of numbers, Messenger of Mathematics, XLV, 1916, 81-84, section (K).
John S. Rutherford, Sublattice enumeration. IV. Equivalence classes of plane sublattices by parent Patterson symmetry and colour lattice group type, Acta Cryst. (2009). A65, 156-163. [See Table 1]. - From _N. J. A. Sloane_, Feb 23 2009
Index entries for sequences related to sums of squares.
Index entries for sequences related to sublattices.
Index entries for "core" sequences.
Index entries for sequences mentioned by Glaisher.

Cf. A000161, A001481, A003881.
Equals 1/4 of A004018. Partial sums give A014200.
Cf. A002175, A008441, A121444, A122856, A122865, A022544, A143574, A000265, A027748, A124010, A025426 (two squares, order does not matter), A120630 (Dirichlet inverse), A101455 (Mobius transform), A000089, A241011.
If one simply reads the table in Glaisher, PLMS 1884, which omits the zero entries, one gets A213408.
Dedekind zeta functions for imaginary quadratic number fields of discriminants -3, -4, -7, -8, -11, -15, -19, -20 are A002324, A002654, A035182, A002325, A035179, A035175, A035171, A035170, respectively.
Dedekind zeta functions for real quadratic number fields of discriminants 5, 8, 12, 13, 17, 21, 24, 28, 29, 33, 37, 40 are A035187, A035185, A035194, A035195, A035199, A035203, A035188, A035210, A035211, A035215, A035219, A035192, respectively.

a002654 n = product $ zipWith f (a027748_row m) (a124010_row m) where
   f p e | p `mod` 4 == 1 = e + 1
         | otherwise      = (e + 1) `mod` 2
   m = a000265 n
-- Reinhard Zumkeller, Mar 18 2013

with(numtheory):
A002654 := proc(n)
    local count1, count3, d;
    count1 := 0:
    count3 := 0:
    for d in numtheory[divisors](n) do
        if d mod 4 = 1 then
            count1 := count1+1
        elif d mod 4 = 3 then
            count3 := count3+1
        fi:
    end do:
    count1-count3;
end proc:
# second Maple program:
a:= n-> add(`if`(d::odd, (-1)^((d-1)/2), 0), d=numtheory[divisors](n)):
seq(a(n), n=1..100);  # Alois P. Heinz, Feb 04 2020

a[n_] := Count[Divisors[n], d_ /; Mod[d, 4] == 1] - Count[Divisors[n], d_ /; Mod[d, 4] == 3]; a/@Range[105] (* Jean-François Alcover, Apr 06 2011, after R. J. Mathar *)
QP = QPochhammer; CoefficientList[(1/q)*(QP[q^2]^10/(QP[q]*QP[q^4])^4-1)/4 + O[q]^100, q] (* Jean-François Alcover, Nov 24 2015 *)
f[2, e_] := 1; f[p_, e_] := If[Mod[p, 4] == 1, e + 1, Mod[e + 1, 2]]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 19 2020 *)
Rest[CoefficientList[Series[EllipticTheta[3, 0, q]^2/4, {q, 0, 100}], q]] (* Vaclav Kotesovec, Mar 10 2023 *)

direuler(p=2,101,1/(1-X)/(1-kronecker(-4,p)*X))

{a(n) = polcoeff( sum(k=1, n, x^k / (1 + x^(2*k)), x * O(x^n)), n)}

{a(n) = sumdiv( n, d, (d%4==1) - (d%4==3))}

{a(n) = local(A); A = x * O(x^n); polcoeff( eta(x^2 + A)^10 / (eta(x + A) * eta(x^4 + A))^4 / 4, n)} \\ Michael Somos, Jun 03 2005

a(n)=my(f=factor(n>>valuation(n,2))); prod(i=1,#f~, if(f[i,1]%4==1, f[i,2]+1, (f[i,2]+1)%2)) \\ Charles R Greathouse IV, Sep 09 2014

my(B=bnfinit(x^2+1)); vector(100,n,#bnfisintnorm(B,n)) \\ Joerg Arndt, Jun 01 2024

from math import prod
from sympy import factorint
def A002654(n): return prod(1 if p == 2 else (e+1 if p % 4 == 1 else (e+1) % 2) for p, e in factorint(n).items()) # Chai Wah Wu, May 09 2022

Dirichlet series: (1-2^(-s))^(-1)*Product (1-p^(-s))^(-2) (p=1 mod 4) * Product (1-p^(-2s))^(-1) (p=3 mod 4) = Dedekind zeta-function of Z[ i ].
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s)+Kronecker(m, p)*p^(-2s))^(-1) for m = -16.
If n=2^k*u*v, where u is product of primes 4m+1, v is product of primes 4m+3, then a(n)=0 unless v is a square, in which case a(n) = number of divisors of u (Jacobi).
Multiplicative with a(p^e) = 1 if p = 2; e+1 if p == 1 (mod 4); (e+1) mod 2 if p == 3 (mod 4). - David W. Wilson, Sep 01 2001
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = (u - v)^2 - (v - w) * (4*w + 1). - Michael Somos, Jul 19 2004
G.f.: Sum_{n>=1} ((-1)^floor(n/2)*x^((n^2+n)/2)/(1+(-x)^n)). - Vladeta Jovovic, Sep 15 2004
Expansion of (eta(q^2)^10 / (eta(q) * eta(q^4))^4 - 1)/4 in powers of q.
G.f.: Sum_{k>0} x^k / (1 + x^(2*k)) = Sum_{k>0} -(-1)^k * x^(2*k - 1) / (1 - x^(2*k - 1)). - Michael Somos, Aug 17 2005
a(4*n + 3) = a(9*n + 3) = a(9*n + 6) = 0. a(9*n) = a(2*n) = a(n). - Michael Somos, Nov 01 2006
a(4*n + 1) = A008441(n). a(3*n + 1) = A122865(n). a(3*n + 2) = A122856(n). a(12*n + 1) = A002175(n). a(12*n + 5) = 2 * A121444(n). 4 * a(n) = A004018(n) unless n=0.
a(n) = Sum_{k=1..n} A010052(k)*A010052(n-k). a(A022544(n)) = 0; a(A001481(n)) > 0.
- Reinhard Zumkeller, Sep 27 2008
a(n) = A001826(n) - A001842(n). - R. J. Mathar, Mar 23 2011
a(n) = Sum_{d|n} A056594(d-1), n >= 1. See the above comment on A056594(d-1) = h(d) of the Niven-Zuckerman reference. - Wolfdieter Lang, Apr 19 2013
Dirichlet g.f.: zeta(s)*beta(s) = zeta(s)*L(chi_2(4),s). - Ralf Stephan, Mar 27 2015
G.f.: (theta_3(x)^2 - 1)/4, where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Apr 17 2018
a(n) = Sum_{ m: m^2|n } A000089(n/m^2). - Andrey Zabolotskiy, May 07 2018
a(n) = A053866(n) + 2 * A025441(n). - Andrey Zabolotskiy, Apr 23 2019
a(n) = Im(Sum_{d|n} i^d). - Ridouane Oudra, Feb 02 2020
a(n) = Sum_{d|n} sin((1/2)*d*Pi). - Ridouane Oudra, Jan 22 2021
Sum_{n>=1} (-1)^n*a(n)/n = Pi*log(2)/4 (Covo, 2010). - Amiram Eldar, Apr 07 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/4 = 0.785398... (A003881). - Amiram Eldar, Oct 11 2022
From Vaclav Kotesovec, Mar 10 2023: (Start)
Sum_{k=1..n} a(k)^2 ~ n * (log(n) + C) / 4, where C = A241011 =
4*gamma - 1 + log(2)/3 - 2*log(Pi) + 8*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.01662154573340811526279685971511542645018417752364748061...
The constant C, published by Ramanujan (1916, formula (22)), 4*gamma - 1 + log(2)/3 - log(Pi) + 4*log(Gamma(3/4)) - 12*Zeta'(2)/Pi^2 = 2.3482276258576... is wrong! (End)

A000161 Number of partitions of n into 2 squares.

Original entry on oeis.org

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

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

A020757 Numbers that are not the sum of two triangular numbers.

Original entry on oeis.org

5, 8, 14, 17, 19, 23, 26, 32, 33, 35, 40, 41, 44, 47, 50, 52, 53, 54, 59, 62, 63, 68, 71, 74, 75, 77, 80, 82, 85, 86, 89, 95, 96, 98, 103, 104, 107, 109, 113, 116, 117, 118, 122, 124, 125, 128, 129, 131, 134, 138, 140, 143, 145, 147, 149, 152, 155, 158, 161, 162, 166, 167

Offset: 1

David W. Wilson

nonn

A052343(a(n)) = 0. - Reinhard Zumkeller, May 15 2006
Numbers of the form (p^(2k+1)s-1)/4, where p is a prime number of the form 4n+3, and s is a number of the form 4m+3 and prime to p, are not expressible as the sum of two triangular numbers. See Satyanarayana (1961), Theorem 2. - Hans J. H. Tuenter, Oct 11 2009
An integer n is in this sequence if and only if at least one 4k+3 prime factor in the canonical form of 4n+1 occurs with an odd exponent. - Ant King, Dec 02 2010
A nonnegative integer n is in this sequence if and only if A000729(n) = 0. - Michael Somos, Feb 13 2011
4*a(n) + 1 are terms of A022544. - XU Pingya, Aug 05 2018 [Actually, k is here if and only if 4*k + 1 is in A022544. - Jianing Song, Feb 09 2021]
Integers m such that the smallest number of triangular numbers which sum to m is 3, hence A061336(a(n)) = 3. - Bernard Schott, Jul 21 2022

			3 = 0 + 3 and 7 = 1 + 6 are not terms, but 8 = 1 + 1 + 6 is a term.

T. D. Noe, Table of n, a(n) for n = 1..10000
John A. Ewell, On Sums of Triangular Numbers and Sums of Squares, The American Mathematical Monthly, Vol. 99, No. 8 (October 1992), pp. 752-757. [From _Ant King_, Dec 02 2010]
U. V. Satyanarayana, On the representation of numbers as sums of triangular numbers, The Mathematical Gazette, 45(351):40-43, February 1961. [From _Hans J. H. Tuenter_, Oct 11 2009]

Complement of A020756.
Cf. A000217, A052343, A022544, A061336.
Numbers k such that the coefficient of x^k in the expansion of Product_{j>=1} (1 - x^j)^m is zero: A090864 (m=1), A213250 (m=2), A014132 (m=3), A302056 (m=4), A302057 (m=5), this sequence (m=6), A322430 (m=8), A322431 (m=10), A322432 (m=14), A322043 (m=15), A322433 (m=26).

a020757 n = a020757_list !! (n-1)
a020757_list = filter ((== 0) . a052343) [0..]
-- Reinhard Zumkeller, Jul 25 2014

data = Reduce[m (m + 1) + n (n + 1) == 2 # && 0 <= m && 0 <= n, {m, n}, Integers] & /@ Range[167]; Position[data, False] // Flatten  (* Ant King, Dec 05 2010 *)
t = Array[PolygonalNumber, 18, 0]; Complement[Range@ 169, Flatten[ Outer[ Plus, t, t]]] (* Robert G. Wilson v, Aug 07 2024 *)

is(n)=my(m9=n%9,f); if(m9==5 || m9==8, return(1)); f=factor(4*n+1); for(i=1,#f~, if(f[i,1]%4==3 && f[i,2]%2, return(1))); 0 \\ Charles R Greathouse IV, Mar 17 2022

cp's OEIS Frontend

A174440 Partial sums of A022544.

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A002145 Primes of the form 4*k + 3.

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A001481 Numbers that are the sum of 2 squares.

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A004767 a(n) = 4*n + 3.

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A004018 Theta series of square lattice (or number of ways of writing n as a sum of 2 squares). Often denoted by r(n) or r_2(n).

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