A117000 -id:A117000 - cp's OEIS frontend

A117005 a(n) = A117000(n) - A117001(n).

1, 2, 2, 2, 2, 2, 2, 2, -1, 2, 2, -4, 2, 2, -4, 2, 2, -4, 2, 2, -4, 2, 2, -4, -3, 2, -4, 2, 2, -14, 2, 2, -4, 2, -8, -4, 2, 2, -4, -8, 2, -4, 2, 2, -14, 2, 2, -4, 9, -8, -4, 2, 2, -4, -8, 16, -4, 2, 2, -14, 2, 2, 10, 2, -8, -4, 2, 2, -4, 6, 2, -4, 2, 2, -14, 2, 16, -4, 2, -8, 5, 2, 2, 10, -8, 2, -4, 2, 2, 4, 16, 2, -4, 2, -8, -4, 2, 16, 14, -8

Offset: 1

N. J. A. Sloane, Apr 15 2006

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H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

Cf. A117000, A117001.

A117006 A117000(n) + A117001(n).

Original entry on oeis.org

1, 0, -6, 0, -10, -6, 14, 0, 15, -10, -22, 0, -26, 14, 20, 0, 34, 18, -38, -10, -28, -22, 46, 0, 45, -26, -36, 14, -58, 30, 62, 0, 44, 34, -56, 18, -74, -38, 52, 0, 82, -28, -86, -22, -42, 46, 94, 0, 105, 50, -68, -26, -106, -36, 88, 0, 76, -58, -118, 30, -122, 62, 102, 0, 104, 44, -134, 34, -92, -70, 142, 18, 146

Offset: 1

N. J. A. Sloane, Apr 15 2006

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H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

A117007 A117000(8n-1) - A117001(8n-1).

Original entry on oeis.org

2, -4, 2, 2, -4, 2, -8, 10, 2, 2, -4, -8, 2, -4, 16, 2, 4, -20, 2, -4, 2, 6, -4, 2, 2, 14, -8, 2, -12, 2, -24, 16, 2, 2, 14, 16, -8, -4, 2, -20, -4, -8, 16, -12, 2, 2, 16, 2, 36, -28, -20, -8, 14, 2, 2, -4, -20, 2, -4, 2, 2, 12, 2, 16, -4, 36, -8, -4, -36, -24, -14, 38, -20, -4, 2, 2, 16, 16, 2, 14, 2, -8, 4, -20, 16, -4, -8, -36, 14, 2

Offset: 1

N. J. A. Sloane, Apr 15 2006

sign

H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

A117008 (A117000(8n-1) - A117001(8n-1))/2.

Original entry on oeis.org

1, -2, 1, 1, -2, 1, -4, 5, 1, 1, -2, -4, 1, -2, 8, 1, 2, -10, 1, -2, 1, 3, -2, 1, 1, 7, -4, 1, -6, 1, -12, 8, 1, 1, 7, 8, -4, -2, 1, -10, -2, -4, 8, -6, 1, 1, 8, 1, 18, -14, -10, -4, 7, 1, 1, -2, -10, 1, -2, 1, 1, 6, 1, 8, -2, 18, -4, -2, -18, -12, -7, 19, -10, -2, 1, 1, 8, 8, 1, 7, 1, -4, 2, -10, 8, -2, -4, -18, 7, 1, 1, -6, 1, 1, 10, -12, 21

Offset: 1

N. J. A. Sloane, Apr 15 2006

sign

H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

A117009 A117000(8n+1) + A117001(8n-1).

Original entry on oeis.org

1, -1, 2, -3, -4, 2, 9, -4, -8, 2, 5, 2, 2, 0, 2, -9, -4, 2, -8, 14, 16, -11, -4, -8, 2, -4, -20, 16, 19, 2, 2, -4, 2, -8, -16, 2, 19, -8, -8, 2, -4, 16, 2, 16, 2, -17, 14, -24, -16, -4, 2, 2, -4, 26, 2, 7, 2, 2, 16, -20, -24, -4, 16, -8, -24, 2, 25, -4, -8, 16, 8, 2, 2, 8, 2, 2, -32, 2, 17, -4, 2, -20, 14, -32, 2, -4, -24, 36, 16, 48

Offset: 0

N. J. A. Sloane, Apr 15 2006

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H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 322.

A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

Original entry on oeis.org

1, 6, 14, 20, 30, 40, 36, 48, 62, 42, 72, 100, 68, 120, 112, 48, 126, 108, 98, 180, 136, 160, 180, 144, 132, 126, 216, 200, 240, 280, 112, 192, 254, 120, 252, 320, 210, 360, 324, 144, 264, 252, 288, 420, 340, 280, 336, 288, 260, 342, 294, 360, 408, 520, 360, 240, 496

Offset: 0

N. J. A. Sloane, Feb 15 2014

Seiichi Manyama, Table of n, a(n) for n = 0..10000
I. J. Zucker, Exact Evaluation of Some New Lattice Sums, Symmetry, 2017, 9(12), 314.

For number of solutions to a^2+b^2+c^2+k*d^2=n for k=1, 2, 3, 4, 5, 6, 7, 8, 12 see A000118, A236928, A236926, A236923, A236930, A236931, A236932, A236927, A236933.

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See A236924.
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G.f.: theta_3(q)^3*theta_3(q^2), where theta_3() is the Jacobi theta function. - Ilya Gutkovskiy, Aug 01 2018

G.f.: 1 + 8*Sum{n >= 1} n*(q^n - q^(3*n) - q^(5*n) + q^(7*n))/(1 - q^(8*n)) - 2*Sum_{n >= 0} (-1)^((n^2+n)/2)*(2*n+1)q^(2*n+1)/(1 - q^(2*n+1)). See Zucker p. 5. Cf. A117000. - Peter Bala, Feb 25 2021

A117001 Sum_{d|n, sqrt(n) < d <= n} Jacobi(2,d)d - Sum_{d|n, 1 <= d < sqrt(n)} Jacobi(2,d)d.

Original entry on oeis.org

0, -1, -4, -1, -6, -4, 6, -1, 8, -6, -12, 2, -14, 6, 12, -1, 16, 11, -20, -6, -12, -12, 22, 2, 24, -14, -16, 6, -30, 22, 30, -1, 24, 16, -24, 11, -38, -20, 28, 4, 40, -12, -44, -12, -14, 22, 46, 2, 48, 29, -32, -14, -54, -16, 48, -8, 40, -30, -60, 22, -62, 30, 46, -1, 56, 24, -68, 16, -44, -38, 70, 11, 72, -38, -28, -20, -96, 28

Offset: 1

N. J. A. Sloane, Apr 15 2006

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H. J. S. Smith, Report on the Theory of Numbers, reprinted in Vol. 1 of his Collected Math. Papers, Chelsea, NY, 1979, see p. 323.

Cf. A117000.

with(numtheory); A117001:=proc(n) local d,t1,t2; t1:=0; t2:=0; for d from 1 to n do if n mod d = 0 then if d^2>n then t1:=t1+jacobi(2,d)*d; fi; if d^2

a[n_] := Sum[Which[Sqrt[n]Jean-François Alcover, Feb 17 2023 *)

A113417 Expansion of phi(x) * phi(-x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

Original entry on oeis.org

1, -2, -4, 8, 7, -10, -12, 8, 18, -18, -16, 24, 21, -20, -28, 32, 20, -32, -36, 24, 42, -42, -28, 48, 57, -36, -52, 40, 36, -58, -60, 56, 48, -66, -48, 72, 74, -42, -80, 80, 61, -82, -72, 56, 90, -96, -64, 72, 98, -70, -100, 104, 64, -106, -108, 72, 114, -96

Offset: 0

Michael Somos, Oct 29 2005

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

			G.f. = 1 - 2*x - 4*x^2 + 8*x^3 + 7*x^4 - 10*x^5 - 12*x^6 + 8*x^7 + 18*x^8 + ...
G.f. = q - 2*q^3 - 4*q^5 + 8*q^7 + 7*q^9 - 10*q^11 - 12*q^13 + 8*q^15 + ...

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113419, A117000, A209940, A258096.

A := Basis( ModularForms( Gamma1(16), 2), 117); A[2] - 2*A[4] - 4*A[6] + 8*A[8] + 7*A[10] - 10*A[12] - 12*A[14]; /* Michael Somos, May 19 2015 */

a[ n_] := If[ n < 0, 0, DivisorSum[ 2 n + 1, KroneckerSymbol[ 2, #] # &]]; (* Michael Somos, May 19 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ x]^2 QPochhammer[ x^8]^2 QPochhammer[ x^2]^3 / QPochhammer[ x^4]^3, {x, 0, n}]; (* Michael Somos, May 19 2015 *)

{a(n) = if( n<0, 0, sumdiv( 2*n + 1, d, d * (d%2) * (-1)^((d + 1) \ 4)))};

{a(n) = my(A, p, e); if( n<0, 0, n = 2*n + 1; A = factor(n); prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 0, p *= kronecker( 2, p); (p^(e+1) - 1) / (p - 1))))};

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

Expansion of q^(-1/2) * (eta(q) * eta(q^8))^2 * (eta(q^2) / eta(q^4))^3 in powers of q.
Euler transform of period 8 sequence [ -2, -5, -2, -2, -2, -5, -2, -4, ...].
a(n) = b(2*n + 1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>=0} a(k) * x^(2*k + 1) = Sum_{k>0} (2*k - 1) * (-1)^[k/2] * x^(2*k - 1) / (1 - x^(4*k - 2)) = x * (Product_{k>0} ((1 - x^(2*k)) * (1 - x^(4*k)) * (1 + x^(8*k)))^2 / (1 + x^(4*k))).
a(n) = (-1)^n * A113419(n) = (-1)^floor(n/2) * A209940(n) = (-1)^(n + floor(n/2)) * A258096(n). - Michael Somos, May 19 2015
a(n) = A117000(2*n + 1).
a(n) = Sum_{d | 2*n + 1} Kronecker(2, d) * d.

A113418 Expansion of (eta(q^2)^7eta(q^4)/(eta(q)eta(q^8))^2-1)/2 in powers of q.

Original entry on oeis.org

1, -1, -2, -1, -4, 2, 8, -1, 7, 4, -10, 2, -12, -8, 8, -1, 18, -7, -18, 4, -16, 10, 24, 2, 21, 12, -20, -8, -28, -8, 32, -1, 20, -18, -32, -7, -36, 18, 24, 4, 42, 16, -42, 10, -28, -24, 48, 2, 57, -21, -36, 12, -52, 20, 40, -8, 36, 28, -58, -8, -60, -32, 56, -1, 48, -20, -66, -18, -48, 32, 72, -7, 74, 36, -42, 18, -80, -24

Offset: 1

Michael Somos, Oct 29 2005

Amiram Eldar, Table of n, a(n) for n = 1..10000

Apart from signs, same as A117000.

A113416(n)=2*a(n) if n>0.

f[p_, e_] := If[1 < Mod[p, 8] < 7, ((-p)^(e+1)-1)/(-p-1), (p^(e+1)-1)/(p-1)]; f[2, e_] := -1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Aug 22 2023 *)

a(n)=if(n<1, 0, -sumdiv(n,d, d*(d%2)*(-1)^(n/d+(d+1)\4)))

{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, p*=kronecker(2,p); (p^(e+1)-1)/(p-1)))))}

a(n) is multiplicative and a(2^e) = -1 if e>0, a(p^e) = (p^(e+1)-1)/(p-1) if p == 1, 7 (mod 8), a(p^e) = ((-p)^(e+1)-1)/(-p-1) if p == 3, 5 (mod 8).
G.f.: Sum_{k>0} (2k-1)*(-1)^[k/2]*x^(2k-1)/(1+x^(2k-1)).
From Amiram Eldar, Jan 28 2024: (Start)
a(n) = (-1)^(n+1) * A117000(n).
Sum_{k=1..n} abs(a(k)) ~ c * n^2, where c = Pi^2/(24*sqrt(2)) = 0.290786... . (End)

A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

Original entry on oeis.org

1, -2, -2, 4, -2, 8, 4, -16, -2, -14, 8, 20, 4, 24, -16, -16, -2, -36, -14, 36, 8, 32, 20, -48, 4, -42, 24, 40, -16, 56, -16, -64, -2, -40, -36, 64, -14, 72, 36, -48, 8, -84, 32, 84, 20, 56, -48, -96, 4, -114, -42, 72, 24, 104, 40, -80, -16, -72, 56, 116, -16

Offset: 0

Michael Somos, Aug 06 2007

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Number 19 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).

			G.f. = 1 - 2*q - 2*q^2 + 4*q^3 - 2*q^4 + 8*q^5 + 4*q^6 - 16*q^7 - 2*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.67).

G. C. Greubel, Table of n, a(n) for n = 0..10000
David Broadhurst and Daniele Dorigoni, Resurgent Lambert series with characters, arXiv:2507.21352 [math.NT], 2025. See p. 37.
Yves Martin, Multiplicative eta-quotients, Trans. Amer. Math. Soc. 348 (1996), no. 12, 4825-4856, see page 4852 Table I.
Michael Somos, Index to Yves Martin's list of 74 multiplicative eta-quotients and their A-numbers
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A113416, A113417, A117000, A124340, A259491.

A := Basis( ModularForms( Gamma1(8), 2), 61); A[1] - 2*A[2] - 2*A[3] + 4*A[4] - 2*A[5]; /* Michael Somos, Jun 28 2015 */

a[ n_] := If[ n < 1, Boole[n == 0], -2 DivisorSum[ n, # KroneckerSymbol[ 2, #] &]]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ QPochhammer[ q]^2 QPochhammer[ q^2] QPochhammer[ q^4]^3 / QPochhammer[ q^8]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, q] EllipticTheta[ 3, 0, q^2] EllipticTheta[ 4, 0, q]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 4, 0, q] EllipticTheta[ 4, 0, q^2] EllipticTheta[ 4, 0, q^4]^2, {q, 0, n}]; (* Michael Somos, Jun 28 2015 *)

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

{a(n) = my(A, p, e); if( n<0, n==0, A = factor(n); -2 * prod( k=1, matsize(A)[1], [p, e] = A[k,]; if( p==2, 1, abs(p%8-4)==3, (p^(e+1) - 1) / (p - 1), ((-p)^(e+1) - 1) / (-p - 1))))};

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

Expansion of phi(q) * phi(q^2) * phi(-q)^2 in powers of q where phi() is a Ramanujan theta function.
Euler transform of period 8 sequence [-2, -3, -2, -6, -2, -3, -2, -4, ...].
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^4 + u^2*v^2 + 2 * u^2*w^2 + 2 * u*v*w * (-u + 2*v - 2*w) - 2 * u*v^3.
a(n) = 2 * b(n) where b() is multiplicative with b(2^e) = 1, b(p^e) = (p^(e+1) - 1) / (p - 1) if p == 1, 7 (mod 8), b(p^e) = ((-p)^(e+1) - 1) / (-p - 1) if p == 3, 5 (mod 8).
a(2*n) = a(n) for all n in Z.
G.f.: 1 - 2* Sum_{k>0} k * x^k / (1 - x^k) * Kronecker(2, k).
G.f.: Product_{k>0} (1 - x^k)^4 * (1 + x^k)^2 * (1 + x^(2*k)) / (1 + x^(4*k))^2.
a(n) = -2 * A117000(n) unless n=0. a(n) = (-1)^n * A113416(n). a(2*n + 1) = - 2 * A113417(n).
G.f. is a period 1 Fourier series which satisfies f(-1 / (8 t)) = 2^(11/2) (t/i)^2 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A124340. - Michael Somos, Jun 28 2015
Convolution square is A259491. - Michael Somos, Jun 28 2015

cp's OEIS Frontend

A117005 a(n) = A117000(n) - A117001(n).

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A117006 A117000(n) + A117001(n).

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A117007 A117000(8*n-1) - A117001(8*n-1).

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A117008 (A117000(8*n-1) - A117001(8*n-1))/2.

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A117009 A117000(8*n+1) + A117001(8*n-1).

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A236928 Number of integer solutions to a^2 + b^2 + c^2 + 2*d^2 = n.

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A117001 Sum_{d|n, sqrt(n) < d <= n} Jacobi(2,d)*d - Sum_{d|n, 1 <= d < sqrt(n)} Jacobi(2,d)*d.

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A113417 Expansion of phi(x) * phi(-x)^2 * psi(x^4) in powers of x where phi(), psi() are Ramanujan theta functions.

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Magma

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A113418 Expansion of (eta(q^2)^7*eta(q^4)/(eta(q)*eta(q^8))^2-1)/2 in powers of q.

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A131999 Expansion of eta(q)^2 * eta(q^2) * eta(q^4)^3 / eta(q^8)^2 in powers of q.

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A117007 A117000(8n-1) - A117001(8n-1).

A117008 (A117000(8n-1) - A117001(8n-1))/2.

A117009 A117000(8n+1) + A117001(8n-1).

A117001 Sum_{d|n, sqrt(n) < d <= n} Jacobi(2,d)d - Sum_{d|n, 1 <= d < sqrt(n)} Jacobi(2,d)d.

A113418 Expansion of (eta(q^2)^7eta(q^4)/(eta(q)eta(q^8))^2-1)/2 in powers of q.