A035203 -id:A035203 - cp's OEIS frontend

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).

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)

A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -3. See Formula section for the general expression. - N. J. A. Sloane, Mar 22 2022
Coefficients in expansion of Dirichlet series Product_p (1 - (Kronecker(m,p) + 1)*p^(-s) + Kronecker(m,p) * p^(-2s))^(-1) for m = -3.
(Number of points of norm n in hexagonal lattice) / 6, n>0.
The hexagonal lattice is the familiar 2-dimensional lattice (A_2) in which each point has 6 neighbors. This is sometimes called the triangular lattice.
The first occurrence of a(n) = 1, 2, 3, 4,... is at n= 1, 7, 49, 91, 2401, 637, ... as tabulated in A343771. - R. J. Mathar, Sep 21 2024

			G.f. = x + x^3 + x^4 + 2*x^7 + x^9 + x^12 + 2*x^13 + x^16 + 2*x^19 + 2*x^21 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, p. 112, first display.
J. W. L. Glaisher, Table of the excess of the number of (3k+1)-divisors of a number over the number of (3k+2)-divisors, Messenger Math., 31 (1901), 64-72.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
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
G. E. Andrews, Three aspects of partitions, Séminaire Lotharingien de Combinatoire, B25f (1990), 1 p.
Hershel M. Farkas, On an arithmetical function, Ramanujan J., 8(3) (2004), 309-315.
Pavel Guerzhoy and Ka Lun Wong, Farkas' identities with quartic characters, arXiv:1905.06506 [math.NT], 2019.
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. [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 below.]
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.
Gabriele Nebe and N. J. A. Sloane, Home page for hexagonal (or triangular) lattice A2.
José Manuel Rodríguez Caballero, Divisors on overlapped intervals and multiplicative functions, arXiv:1709.09621 [math.NT], 2017.
J. S. Rutherford, Generating functions for the cage isomers of the C_{20n} icosahedral fullerenes, J. Mathematical Chem., 14 (1993), 385-390. [From _N. J. A. Sloane_, Mar 12 2009]
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
N. J. A. Sloane, Maple code for Dedekind zeta functions of quadratic number fields.

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.
Cf. A004016, A035019, A073010, A145377, A293899, A000086, A003136, A034020, A145394.

a002324 n = a001817 n - a001822 n  -- Reinhard Zumkeller, Nov 26 2011

A002324 := proc(n)
    local a,pe,p,e;
    a :=1 ;
    for pe in ifactors(n)[2] do
        p := op(1,pe) ;
        e := op(2,pe) ;
        if p = 3 then
            ;
        elif modp(p,3) = 1 then
            a := a*(e+1) ;
        else
            a := a*(1+(-1)^e)/2 ;
        end if;
    end do:
    a ;
end proc:
seq(A002324(n),n=1..100) ; # R. J. Mathar, Sep 21 2024

dn12[n_]:=Module[{dn=Divisors[n]},Count[dn,?(Mod[#,3]==1&)]-Count[ dn,?(Mod[#,3]==2&)]]; dn12/@Range[120]  (* Harvey P. Dale, Apr 26 2011 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -3, #] &]]; (* Michael Somos, Aug 24 2014 *)
Table[DirichletConvolve[DirichletCharacter[3,2,m],1,m,n],{n,1,30}] (* Steven Foster Clark, May 29 2019 *)
f[3, p_] := 1; f[p_, e_] := If[Mod[p, 3] == 1, e+1, (1+(-1)^e)/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 17 2020 *)

{a(n) = if( n<0, 0, polcoeff( sum(k=1, n, x^k / (1 + x^k + x^(2*k)), x * O(x^n)), n))}; \\ Michael Somos

{a(n) = if( n<1, 0, sumdiv(n, d, (d%3==1) - (d%3==2)))};

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], if( p=A[k,1], e=A[k,2]; if( p==3, 1, if( p%3==1, e+1, !(e%2))))))}; \\ Michael Somos, May 20 2005

{a(n) = if( n<1, 0, qfrep([2,1; 1,2], n, 1)[n] / 3)}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker(-3, p)*X))[n])}; \\ Michael Somos, Jun 05 2005

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

from math import prod
from sympy import factorint
def A002324(n): return prod(e+1 if p%3==1 else int(not e&1) for p, e in factorint(n).items() if p != 3) # Chai Wah Wu, Nov 17 2022

From N. J. A. Sloane, Mar 22 2022 (Start):
The Dedekind zeta function DZ_K(s) for a quadratic field K of discriminant D is as follows.
Here m is defined by K = Q(sqrt(m)) (so m=D/4 if D is a multiple of 4, otherwise m=D).
DZ_K(s) is the product of three terms:
(a) Product_{odd primes p | D} 1/(1-1/p^s)
(b) Product_{odd primes p such that (D|p) = -1} 1/(1-1/p^(2s))
(c) Product_{odd primes p such that (D|p) = 1} 1/(1-1/p^s)^2
and if m is
   0,1,2,3,4,5,6,7 mod 8, the prime 2 is to be included in term
   -,c,a,a,-,b,a,a, respectively.
For Maple (and PARI) implementations, see link. (End)
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 - 3*v^2 + 4*w^2 - 2*u*w + w - v. - Michael Somos, Jul 20 2004
Has a nice Dirichlet series expansion, see PARI line.
G.f.: Sum_{k>0} x^k/(1+x^k+x^(2*k)). - Vladeta Jovovic, Dec 16 2002
a(3*n + 2) = 0, a(3*n) = a(n), a(3*n + 1) = A033687(n). - Michael Somos, Apr 04 2003
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^3), A(x^6)) where f(u1, u2, u3, u6) = (u1 - u3)*(u3 - u6) - (u2 - u6)^2. - Michael Somos, May 20 2005
Multiplicative with a(3^e) = 1, a(p^e) = e+1 if p == 1 (mod 3), a(p^e) = (1+(-1)^e)/2 if p == 2 (mod 3). - Michael Somos, May 20 2005
G.f.: Sum_{k>0} x^(3*k - 2) / (1 - x^(3*k - 2)) - x^(3*k - 1)  / (1 - x^(3*k - 1)). - Michael Somos, Nov 02 2005
G.f.: Sum_{n >= 1} q^(n^2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009
a(n) = A001817(n) - A001822(n). - R. J. Mathar, Mar 31 2011
A004016(n) = 6*a(n) unless n=0.
Dirichlet g.f.: zeta(s)*L(chi_2(3),s), with chi_2(3) the nontrivial Dirichlet character modulo 3 (A102283). - Ralf Stephan, Mar 27 2015
From Andrey Zabolotskiy, May 07 2018: (Start)
a(n) = Sum_{ m: m^2|n } A000086(n/m^2).
a(A003136(m)) > 0, a(A034020(m)) = 0 for all m. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(3*sqrt(3)) = 0.604599... (A073010). - Amiram Eldar, Oct 11 2022

More terms from David Radcliffe

Somos D.g.f. replaced with correct version by Ralf Stephan, Mar 27 2015

A002325 Glaisher's J numbers.

Original entry on oeis.org

1, 1, 2, 1, 0, 2, 0, 1, 3, 0, 2, 2, 0, 0, 0, 1, 2, 3, 2, 0, 0, 2, 0, 2, 1, 0, 4, 0, 0, 0, 0, 1, 4, 2, 0, 3, 0, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 2, 1, 1, 4, 0, 0, 4, 0, 0, 4, 0, 2, 0, 0, 0, 0, 1, 0, 4, 2, 2, 0, 0, 0, 3, 2, 0, 2, 2, 0, 0, 0, 0, 5, 2, 2, 0, 0, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 2, 2, 1, 6, 1, 0, 4, 0, 0, 0

Offset: 1

N. J. A. Sloane

Number of integer solutions to the equation x^2 + 2*y^2 = n when (-x, -y) and (x, y) are counted as the same solution.
For n nonzero, a(n) is nonzero if and only if n is in A002479. - Michael Somos, Dec 15 2011
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			x + x^2 + 2*x^3 + x^4 + 2*x^6 + x^8 + 3*x^9 + 2*x^11 + 2*x^12 + x^16 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 114 Entry 8(iii).
L. E. Dickson, History of the Theory of Numbers. Carnegie Institute Public. 256, Washington, DC, Vol. 1, 1919; Vol. 2, 1920; Vol. 3, 1923, see vol. 3, p. 19.
N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 78, Eq. (32.24).
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91.
D. H. Lehmer, Guide to Tables in the Theory of Numbers. Bulletin No. 105, National Research Council, Washington, DC, 1941, pp. 7-10.
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
J. W. L. Glaisher, Table of the excess of the number of (8k+1)- and (8k+3)-divisors of a number over the number of (8k+5)- and (8k+7)-divisors, Messenger Math., 31 (1901), 82-91. [Incomplete annotated scanned copy]
Michael D. Hirschhorn, The number of representations of a number by various forms, Discrete Mathematics 298 (2005), 205-211.
Christian Kassel and Christophe Reutenauer, Pairs of intertwined integer sequences, arXiv:2507.15780 [math.NT], 2025. See p. 5.
N. J. A. Sloane, Transforms.
Index entries for sequences related to Glaisher's numbers.

Cf. A033715, A093954.
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.

S:= series( (JacobiTheta3(0,q)*JacobiTheta3(0,q^2)-1)/2, q, 1001):
seq(coeff(S,q,j), j=1..1000); # Robert Israel, Dec 01 2015

a[n_] := Total[ KroneckerSymbol[-8, #] & /@ Divisors[n]]; Table[a[n], {n, 1, 105}] (* Jean-François Alcover, Nov 25 2011, after Michael Somos *)
QP = QPochhammer; s = ((QP[q^2]^3*QP[q^4]^3)/(QP[q]^2*QP[q^8]^2)-1)/(2q) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

a(n) = if( n<1, 0, issquare(n)-issquare(2*n) + 2*sum(i=1,sqrtint(n\2), issquare(n-2*i^2)))

{a(n) = if( n<1, 0, qfrep([ 1, 0; 0, 2],n)[n])} \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / (1 - X) / (1 - kronecker( -2, p) * X))[n])} \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -2, d)))} \\ Michael Somos, Aug 23 2005

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p=A[k, 1], e=A[k, 2]; if( p==2, 1, if( p%8<4, e+1, !(e%2))))))} \\ Michael Somos, Oct 23 2006

{a(n) = local(A); if( n<1, 0, A = x * O(x^n); polcoeff( eta(x + A)^-2 * eta(x^2 + A)^3 * eta(x^4 + A)^3 * eta(x^8 + A)^-2, n) / 2)}

a(n) = my(f=factor(n>>valuation(n,2)), e); prod(i=1, #f~, e=f[i, 2]; if( f[i, 1]%8<4, e+1, 1 - e%2)) \\ Charles R Greathouse IV, Sep 09 2014

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m, p)+1)*p^(-s) + Kronecker(m, p)*p^(-2s))^(-1) for m = -2.
Moebius transform is period 8 sequence [ 1, 0, 1, 0, -1, 0, -1, 0, ...]. - Michael Somos, Aug 23 2005
G.f.: (theta_3(q) * theta_3(q^2) - 1) / 2 = Sum_{k>0} Kronecker( -2, n) * x^k / (1 - x^k) = Sum_{k>0} (x^k + x^(3*k)) / (1 + x^(4*k)).
Multiplicative with a(2^e) = 1, a(p^e) = e+1 if p == 1, 3 (mod 8), a(p^e) = (1+(-1)^e)/2 if p == 5, 7 (mod 8). - Michael Somos, Oct 23 2006
A033715(n) = 2 * a(n) unless n=0.
a(n) = A188169(n) + A188170(n) - A188171(n) - A188172(n) [Hirschhorn]. - R. J. Mathar, Mar 23 2011
G.f.: A(x) = 2*(1+x^2)/(G(0)-2*x*(1+x^2)); G(k) = 1+x+x^(2*k)*(1+x^3+x^(2*k+1)+x^(2*k+4)+x^(4*k+3)+x^(4*k+4)) - x*(1+x^(2*k))*(1+x^(2*k+4))*(1+x^(4*k+4))^2/G(k+1); (continued fraction). - Sergei N. Gladkovskii, Jan 03 2012
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/(2*sqrt(2)) = 1.110720... (A093954). - Amiram Eldar, Oct 11 2022

A035187 Sum over divisors d of n of Kronecker symbol (5|d).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Let tau be the golden ratio (1+sqrt(5))/2; let zetaQ(tau)(s)=sum(1/(Z(tau):a)^s) the Dedekind zeta function where a runs through the nonzero ideals of Z(tau) and where (Z(tau):a) is the norm of a; then zetaQ(tau)(s)=sum(n>=1,a(n)/n^s). - Benoit Cloitre, Dec 29 2002
First occurrence of k beginning at zero, or 0 if not yet known: 2, 1, 11, 121, 209, 14641, 2299, 1771561, 6061, 43681, 278179, 0, 66671, 0, 33659659, 5285401, 187891, 0, 1266749, 0, 8067191, 639533521, 0, 0, 2066801, 0, 0, 36735721, 976130111, 0, 153276629, 0, 7703531, 0, 0, 0, 39269219, 0, 0, 0, 250082921, 0, 0, 0, 0, 0, 0, 0, 84738841, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 454508329, ..., .
If k is prime, the 0 above can be replaced by the smallest p^(k-1) with p a prime == {1,4} (mod 5), which is p=11. This follows from the multiplicative formula. - R. J. Mathar, Apr 02 2011
The terms often equal A001157(n) mod 5; the exceptions are at n = 2299, 3509, 3751, 3971, 4961, 6061, 6479, ... - R. J. Mathar, Apr 02 2011
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 5. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + x^4 + x^5 + x^9 + 2*x^11 + x^16 + 2*x^19 + x^20 + x^25 + 2*x^29 + ...

G. C. Greubel, Table of n, a(n) for n = 1..5000
M. Baake, Algebra, Combinatorics and Number Theory.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Cf. A031363 (for indices of nonzero terms), A078428.
Cf. A001622, A086466.
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.

A035187 := proc(n) local f,p; f := ifactors(n)[2] ; if nops(f) = 1 then p := op(1,f) ; if op(1,p) = 5 then 1; elif op(1,p) mod 5 in {1,4} then op(2,p)+1 ; else (1+(-1)^op(2,p))/2 ; end if; else mul(procname(op(1,p)^op(2,p) ),p=f) ; end if;
end proc: # R. J. Mathar, Apr 02 2011

f[n_] := Plus @@ (KroneckerSymbol[5, #] & /@ Divisors@ n); Array[f, 105] (* Robert G. Wilson v *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 5, #] &]]; (* Michael Somos, Jun 12 2014 *)

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( 5, p) * X))[n])}; \\ Michael Somos, Jun 06 2005

{a(n) = local(A, p, e); if( n<1, 0, A = factor(n); prod( k=1, matsize(A)[1], if( p = A[k,1], e = A[k,2]; if( p==5, 1, if((p%5==1) || (p%5==4), e+1, !(e%2))))))}; \\ Michael Somos, Jun 06 2005

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( 5, d) ) )}; \\ Michael Somos, Oct 29 2005

Dirichlet g.f.: Product_p ( (1 - p^(-s)) (1 - Kronecker( 5, p)*p^(-s)) )^(-1).
Sum_{k=1..n} a(k) is asymptotic to c*n where c=2*log(tau)/sqrt(5) (A086466).
Multiplicative with a(5^e) = 1, a(p^e) = e+1 if p == 1, 4 (mod 5), a(p^e) = (1+(-1)^e)/2 if p == 2, 3 (mod 5). - Michael Somos, Jun 06 2005
Moebius transform is period 5 sequence A080891. - Michael Somos, Oct 29 2005
q-series for a(n): Sum_{n >= 1} -(-1)^nq^(n(n+1)/2)(1-q)(1-q^2)...(1-q^(n-1))/((1-q^(n+1))(1-q^(n+2))...(1-q^(2n))). - Jeremy Lovejoy, Jun 12 2009

A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = 2.
Let zetaQ(sqrt(2))(s) = Sum (1/(Z(sqrt(2)):A)^s), a Dedekind zeta function, where A runs through the nonzero ideals of Z(sqrt(2)) and where (Z(sqrt(2)):A) is the norm of A; then zetaQ(sqrt(2))(s) = Sum_{n>=1}, a(n)/n^s); Sum{k=1..n} a(k) is asymptotic to c*n where c = log(1 + sqrt(2))/sqrt(2). - Benoit Cloitre, Jan 01 2003
Inverse Moebius transform of A091337.
a(n) is the number of solutions to the equation n = x^2 - 2*y^2 in integers where -x < 2*y <= x. [Uspensky and Heaslet] - Michael Somos, Feb 17 2020
Coefficients of Dedekind zeta function for the quadratic number field of discriminant 8. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + x^2 + x^4 + 2*x^7 + x^8 + x^9 + 2*x^14 + x^16 + 2*x^17 + x^18 + ...
a(7) = 2 because 7 = 3^2 - 2*(+1)^2 = 3^2 - 2*(-1)^2. - _Michael Somos_, Feb 17 2020

J. V. Uspensky and M. A. Heaslet, Elementary Number Theory, McGraw-Hill, NY, 1939, p. 368.

G. C. Greubel, Table of n, a(n) for n = 1..10000
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, arXiv:math/9904028 [math.MG], 1999.
M. Baake and R. V. Moody, Similarity submodules and root systems in four dimensions, Canad. J. Math. 51 (1999), 1258-1276.

Cf. A035251, A078462, A188169, A188170, A188171, A188172, A259829.
Cf. A002193, A091648.
Moebius transform gives A091337.
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.

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := SeriesCoefficient[ Sum[ x^k (1 - x^(2 k)) / (1 + x^(4 k)), {k, n}], {x, 0, n}]; (* Michael Somos, Jul 06 2015 *)
a[ n_] := If[ n < 1, 0, Times @@ (Which[ # <= 2, 1, Mod[#, 8] > 1 && Mod[#, 8] < 7, 1 - Mod[#2, 2], True, #2 + 1] & @@@ FactorInteger @ n)]; (* Michael Somos, Jul 06 2015 *)

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker(2, d)))};

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker(2, p)*X)))[n])};

{a(n) = my(A, p, e); if( n<1, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k, ]; if(p==2, 1, p%8>1 && p%8<7, !(e%2), e+1)))}; \\ Michael Somos, Aug 17 2006

{a(n) = if( n<1, 0, polcoeff( sum(k=1, n, x^k * (1 - x^(2*k)) / (1 + x^(4*k)), x * O(x^n)), n))}; \\ Michael Somos, Jul 06 2015

G.f.: Sum_{k>0} x^k * (1 - x^(2*k)) / (1 + x^(4*k)).
-(-1)^(n*(n-1)/2)*a(n) = Sum_{n >= 1} (-1)^n * q^(n*(n+1)/2)*(1-q)*(1-q^2)*...*(1-q^(n-1))/ ((1+q)*(1+q^2)*...*(1+q^n)). - Jeremy Lovejoy, Jun 12 2009
a(n) = (-1)^floor(n/2) * A259829(n). - Michael Somos, Jul 06 2015
a(n) is multiplicative with a(2^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5 (mod 8), a(p^e) = e + 1 if p == 1, 7 (mod 8). - Jianing Song, Sep 07 2018
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = log(sqrt(2)+1)/sqrt(2) = A091648/A002193 = 0.623225... . - Amiram Eldar, Oct 11 2022

A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^2 + 5*v^2 + 4*w^2 - 8*v*w - 4*u*v + 2*u*w + v - w. - Michael Somos, Jul 21 2004
Half of the number of integer solutions to x^2 + x*y + 2*y^2 = n. - Michael Somos, Jun 05 2005
Inverse Moebius transform of A175629. - Jianing Song, Sep 07 2018
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -7. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + 2*x^2 + 3*x^4 + x^7 + 4*x^8 + x^9 + 2*x^11 + 2*x^14 + 5*x^16 + ...

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A002652, A326919.
Moebius transform gives A175629.
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.

A := Basis( ModularForms( Gamma1(14), 1), 106); B := (-1 + A[1] + 2*A[2] + 4*A[3] + 6*A[5]) / 2; B; // Michael Somos, Jun 10 2015

a[ n_] := If[ n < 1, 0, Sum[ KroneckerSymbol[ -7, d], { d, Divisors[ n]}]]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, Length @ FindInstance[ n == x^2 + x y + 2 y^2, {x, y}, Integers, 10^9] / 2]; (* Michael Somos, Jan 23 2014 *)
a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -7, #] &]]; (* Michael Somos, Jun 10 2015 *)

{a(n) = my(A, p, e); if( n<0, 0, A = factor(n); prod(k=1, matsize(A)[1], [p, e] = A[k,]; [ !(e%2), 1, e+1] [kronecker( -7, p) + 2]))}; \\ Michael Somos, May 28 2005

{a(n) = if( n<1, 0, qfrep([ 2, 1; 1, 4], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / ((1 - X) * (1 - kronecker( -7, p)*X)))[n])}; \\ Michael Somos, Jun 05 2005

a(n) is multiplicative with a(7^e) = 1, a(p^e) = e + 1 if p == 1, 2, 4 (mod 7), a(p^e) = (1 + (-1)^e) / 2 if p == 3, 5, 6 (mod 7). - Michael Somos, May 28 2005
2 * a(n) = A002652(n) unless n = 0.
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(7) =  1.187410... (A326919). - Amiram Eldar, Oct 11 2022

A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -20.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 2, 1, 3, 1, 0, 2, 0, 2, 2, 1, 0, 3, 0, 1, 4, 0, 2, 2, 1, 0, 4, 2, 2, 2, 0, 1, 0, 0, 2, 3, 0, 0, 0, 1, 2, 4, 2, 0, 3, 2, 2, 2, 3, 1, 0, 0, 0, 4, 0, 2, 0, 2, 0, 2, 2, 0, 6, 1, 0, 0, 2, 0, 4, 2, 0, 3, 0, 0, 2, 0, 0, 0, 0, 1, 5, 2, 2, 4, 0, 2, 4, 0, 2, 3, 0, 2, 0, 2, 0, 2, 0, 3, 0, 1, 2, 0, 2, 0, 4

Offset: 1

N. J. A. Sloane

Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -20. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			q + q^2 + 2*q^3 + q^4 + q^5 + 2*q^6 + 2*q^7 + q^8 + 3*q^9 + q^10 + ...

B. C. Berndt, Ramanujan's Notebooks Part III, Springer-Verlag, see p. 253.

G. C. Greubel, Table of n, a(n) for n = 1..10000
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

Cf. A028586, A033718, A033764, A111949, A124233, A129390.
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.

QP = QPochhammer; s = (1/q) * (QP[q^2]*QP[q^4]*QP[q^5]*(QP[q^10] / (QP[q]* QP[q^20]))-1) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Dec 04 2015 *)
a[n_] := If[n < 0, 0, DivisorSum[ n, KroneckerSymbol[-20, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Dec 12 2017 *)

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

{a(n) = if( n<1, 0, sumdiv(n, d, kronecker( -20, d)))} \\ Michael Somos, Sep 10 2005

{a(n) = if( n<1, 0, direuler( p=2, n, 1 / (1 - X) / (1 - kronecker( -20, p) * X) )[n])} \\ Michael Somos, Sep 10 2005

{a(n) = if( n<1, 0, qfrep([1, 0; 0, 5], n)[n] + qfrep([2, 1; 1, 3], n)[n])} \\ Michael Somos, Oct 21 2006

Multiplicative with a(2^e) = a(5^e) = 1, a(p^e) = e+1 if p == 1, 3, 7, 9 (mod 20), a(p^e) = (1+(-1)^e)/2 if p == 11, 13, 17, 19 (mod 20). - Michael Somos, Sep 10 2005
G.f.: Sum_{k>0} x^k * (1 + x^(2*k)) * (1 + x^(6*k)) / (1 + x^(10*k)). - Michael Somos, Sep 10 2005
a(2*n) = a(5*n) = a(n), a(20*n + 11) = a(20*n + 13) = a(20*n + 17) = a(20*n + 19) = 0.
Moebius transform is period 20 sequence [ 1, 0, 1, 0, 0, 0, 1, 0, 1, 0, -1, 0, -1, 0, 0, 0, -1, 0, -1, 0, ...]. - Michael Somos, Oct 21 2006
Expansion of -1 + (phi(q) * phi(q^5) + phi(q^2) * phi(q^10) + 4 * q^3 * psi(q^4)* psi(q^20)) / 2 in powers of q where phi(), psi() are Ramanujan theta functions.
2*a(n) = A028586(n) + A033718(n) if n>0. - Michael Somos, Oct 21 2006
a(n) = A124233(n) unless n=0. a(n) = |A111949(n)|. a(2*n + 1) = A129390(n). a(4*n + 3) = 2 * A033764(n).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(5) = 1.404962... . - Amiram Eldar, Oct 11 2022

A035179 a(n) = Sum_{d|n} Kronecker(-11, d).

Original entry on oeis.org

1, 0, 2, 1, 2, 0, 0, 0, 3, 0, 1, 2, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, 2, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015
Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005
From Jianing Song, Sep 07 2018: (Start)
Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.
Inverse Moebius transform of A011582. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -11. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...

Henry McKean and Victor Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032).

G. C. Greubel, Table of n, a(n) for n = 1..5000

Cf. A028609, A065099, A129522, A138661.
Moebius transform gives A011582.
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.

A := Basis( ModularForms( Gamma1(11), 1), 88); B := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; // Michael Somos, Jun 07 2015

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)

{a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; \\ Michael Somos, Jun 05 2005

{a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};

a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007
Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007
G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007
A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(11) = 0.947225... . - Amiram Eldar, Oct 11 2022

A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

From Jianing Song, Sep 07 2018: (Start)
Half of the number of integer solutions to x^2 + x*y + 5*y^2 = n.
Inverse Moebius transform of A011585. (End)
Coefficients of Dedekind zeta function for the quadratic number field of discriminant -19. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

G. C. Greubel, Table of n, a(n) for n = 1..10000

Cf. A028641.
Moebius transform gives A011585.
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.

a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-19, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -19; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

From Jianing Song, Sep 07 2018: (Start)
a(n) is multiplicative with a(19^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if Kronecker(-19, p) = -1, a(p^e) = e + 1 if Kronecker(-19, p) = 1.
G.f.: Sum_{k>0} Kronecker(-19, k) * x^k / (1 - x^k).
A028641(n) = 2 * a(n) unless n = 0.
(End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Pi/sqrt(19) = 0.720730... . - Amiram Eldar, Oct 11 2022

A035175 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -15.

Original entry on oeis.org

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

Offset: 1

N. J. A. Sloane

Coefficients of Dedekind zeta function for the quadratic number field of discriminant -15. See A002324 for formula and Maple code. - N. J. A. Sloane, Mar 22 2022

			q + 2*q^2 + q^3 + 3*q^4 + q^5 + 2*q^6 + 4*q^8 + q^9 + 2*q^10 +...

G. C. Greubel, Table of n, a(n) for n = 1..10000

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.

QP = QPochhammer; s = (QP[q^3]*QP[q^5])^2/(QP[q]*QP[q^15])/q - 1/q + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 30 2015, adapted from PARI *)
a[n_] := If[n < 0, 0, DivisorSum[n, KroneckerSymbol[-15, #] &]]; Table[a[n], {n, 1, 100}] (* G. C. Greubel, Jul 17 2018 *)

m = -15; direuler(p=2,101,1/(1-(kronecker(m,p)*(X-X^2))-X))

{a(n)=if(n<1, 0, sumdiv(n, d, kronecker(-15,d)))} \\ Michael Somos, Aug 25 2006

{a(n)=local(A, p, e); if(n<1, 0, A=factor(n); prod(k=1, matsize(A)[1], if(p=A[k, 1], e=A[k, 2]; if(p==3||p==5, 1, if((p%15)!=2^valuation(p%15,2), (e+1)%2, (e+1))))))} \\ Michael Somos, Aug 25 2006

{a(n)=if(n<1, 0, (qfrep([2, 1;1, 8],n, 1)+qfrep([4, 1;1, 4], n, 1))[n])} \\ Michael Somos, Aug 25 2006

{a(n)=local(A); if(n<1, 0, A=x*O(x^n); polcoeff( eta(x^3+A)^2*eta(x^5+A)^2/eta(x+A)/eta(x^15+A), n))} \\ Michael Somos, Aug 25 2006

From Michael Somos, Aug 25 2006: (Start)
Expansion of -1 + (eta(q^3) * eta(q^5))^2 / (eta(q) * eta(q^15)) in powers of q.
Euler transform of period 15 sequence [ 1, 1, -1, 1, -1, -1, 1, 1, -1, -1, 1, -1, 1, 1, -2, ...]. if a(0)=1.
Moebius transform is period 15 sequence [ 1, 1, 0, 1, 0, 0, -1, 1, 0, 0, -1, 0, -1, -1, 0, ...].
Given g.f. A(x), then B(x) = 1 + A(x) satisfies 0 = f(B(x), B(x^2), B(x^4)) where f(u, v, w) = -v^3 + 4*u*v*w - 2*u*w^2 - u^2*w.
G.f.: -1 + x * Product_{k>0} ((1 - x^(3*k)) * (1 - x^(5*k)))^2 / ((1 - x^k) * (1 - x^(15*k))).
G.f.: -1 + (1/2) * (Sum_{n,m} x^(n^2 + n*m + 4*m^2) + x^(2*n^2 + n*m + 2*m^2)).
a(n) is multiplicative with a(3^e) = a(5^e) = 1, a(p^e) = (1+(-1)^e)/2 if p == 7, 11, 13, 14 (mod 15), a(p^e) = e+1 if p == 1, 2, 4, 8 (mod 15).
a(15*n + 7) = a(15*n + 11) = a(15*n + 13) = a(15*n + 14) = 0.
a(3*n) = a(n). a(n) = |A106406(n)| unless n=0. a(n) = A123864(n) unless n=0. (End)
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = 2*Pi/sqrt(15) = 1.622311... . - Amiram Eldar, Oct 11 2022

cp's OEIS Frontend

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).

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A002324 Number of divisors of n == 1 (mod 3) minus number of divisors of n == 2 (mod 3).

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A002325 Glaisher's J numbers.

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A035187 Sum over divisors d of n of Kronecker symbol (5|d).

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A035185 Number of divisors of n == 1 or 7 (mod 8) minus number of divisors of n == 3 or 5 (mod 8).

A035182 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -7.

A035170 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s)+Kronecker(m,p)p^(-2s))^(-1) for m = -20.

A035171 Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)p^(-s) + Kronecker(m,p)p^(-2s))^(-1) for m = -19.