A050470 -id:A050470 - cp's OEIS frontend

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

1, 1024, 59048, 1048576, 9765626, 60465152, 282475248, 1073741824, 3486725353, 10000001024, 25937424600, 61916315648, 137858491850, 289254653952, 576640684048, 1099511627776, 2015993900450, 3570406761472, 6131066257800, 10240001048576

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, this sequence, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^10 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(10*e+10) - s[p]^(e+1))/(p^10 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n, d, if(bittest(n\d,0),(2-n\d%4)*d^10)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^10*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(10*e+10) - A101455(p)^(e+1))/(p^10 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^11 / 11, where c = beta(11) = 50521*Pi^11/14863564800 = 0.999994374973... and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^10*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

Original entry on oeis.org

1, 2048, 177146, 4194304, 48828126, 362795008, 1977326742, 8589934592, 31380882463, 100000002048, 285311670610, 743004176384, 1792160394038, 4049565167616, 8649707208396, 17592186044416, 34271896307634, 64268047284224, 116490258898218

Offset: 1

N. J. A. Sloane, Nov 24 2018

Seiichi Manyama, Table of n, a(n) for n = 1..10000
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 (see p. 4 and p. 8).
Index entries for sequences mentioned by Glaisher.

Cf. A101455.
Cf. A321807 - A321836 for related sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, this sequence, A321836.

s[n_,r_] := DivisorSum[n, #^11 &, Mod[n/#,4]==r &]; a[n_] := s[n,1] - s[n,3]; Array[a, 30] (* Amiram Eldar, Nov 26 2018 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(11*e+11) - s[p]^(e+1))/(p^11 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( a(n)=sumdiv(n,d,if(bittest(n\d,0),(2-n\d%4)*d^11)), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^11*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(11*e+11) - A101455(p)^(e+1))/(p^11 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^12 / 12, where c = beta(12) = 0.99999812235..., and beta is the Dirichlet beta function. (End)
a(n) = Sum_{d|n} (n/d)^11*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A050461 a(n) = Sum_{d|n, n/d=1 mod 4} d^2.

Original entry on oeis.org

1, 4, 9, 16, 26, 36, 49, 64, 82, 104, 121, 144, 170, 196, 234, 256, 290, 328, 361, 416, 442, 484, 529, 576, 651, 680, 738, 784, 842, 936, 961, 1024, 1090, 1160, 1274, 1312, 1370, 1444, 1530, 1664, 1682, 1768, 1849, 1936, 2132, 2116, 2209

Offset: 1

N. J. A. Sloane, Dec 23 1999

Not multiplicative: a(3)*a(7) <> a(21), for example. - R. J. Mathar, Dec 20 2011

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A050465, A050470, A076577, A027750, A002117.

Cf. A050460, A050462, A050463.

a050461 n = sum [d ^ 2 | d <- a027750_row n, mod (div n d) 4 == 1]
-- Reinhard Zumkeller, Mar 06 2012

A050461 := proc(n)
        a := 0 ;
        for d in numtheory[divisors](n) do
                if (n/d) mod 4 = 1 then
                        a := a+d^2 ;
                end if;
        end do:
        a;
end proc:
seq(A050461(n),n=1..40) ; # R. J. Mathar, Dec 20 2011

a[n_] := DivisorSum[n, Boole[Mod[n/#, 4] == 1]*#^2&]; Array[a, 50] (* Jean-François Alcover, Feb 12 2018 *)

a(n) = sumdiv(n, d, (n/d % 4 == 1) * d^2); \\ Amiram Eldar, Nov 05 2023

a(n) = A050470(n) + A050465(n). - Reinhard Zumkeller, Mar 06 2012
From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A076577(n) - A050465(n).
a(n) = (A050470(n) + A076577(n))/2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = Pi^3/64 + 7*zeta(3)/16 = 1.010372968262... . (End)

A050465 a(n) = Sum_{d|n, n/d=3 mod 4} d^2.

Original entry on oeis.org

0, 0, 1, 0, 0, 4, 1, 0, 9, 0, 1, 16, 0, 4, 26, 0, 0, 36, 1, 0, 58, 4, 1, 64, 0, 0, 82, 16, 0, 104, 1, 0, 130, 0, 26, 144, 0, 4, 170, 0, 0, 232, 1, 16, 234, 4, 1, 256, 49, 0, 290, 0, 0, 328, 26, 64, 370, 0, 1, 416, 0, 4, 523, 0, 0, 520, 1, 0, 538, 104, 1, 576, 0

Offset: 1

N. J. A. Sloane, Dec 23 1999

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A002117, A027750, A050461, A050470, A076577.

Cf. A050464, A050466, A050467.

a050465 n = sum [d ^ 2 | d <- a027750_row n, mod (div n d) 4 == 3]
-- Reinhard Zumkeller, Mar 06 2012

a[n_] := DivisorSum[n, #^2 &, Mod[n/#, 4] == 3 &]; Array[a, 100] (* Amiram Eldar, Nov 05 2023 *)

a(n) = sumdiv(n, d, (n/d % 4 == 3) * d^2); \\ Amiram Eldar, Nov 05 2023

a(n) = A050461(n) - A050470(n). - Reinhard Zumkeller, Mar 06 2012
From Amiram Eldar, Nov 05 2023: (Start)
a(n) = A076577(n) - A050461(n).
a(n) = (A076577(n) - A050470(n))/2.
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = 7*zeta(3)/16 - Pi^3/64 = 0.041426822002... . (End)

Offset fixed by Reinhard Zumkeller, Mar 06 2012

A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

Original entry on oeis.org

1, 1, 1, 1, 2, 0, 1, 4, 2, 1, 1, 8, 8, 4, 2, 1, 16, 26, 16, 6, 0, 1, 32, 80, 64, 26, 4, 0, 1, 64, 242, 256, 126, 32, 6, 1, 1, 128, 728, 1024, 626, 208, 48, 8, 1, 1, 256, 2186, 4096, 3126, 1280, 342, 64, 7, 2, 1, 512, 6560, 16384, 15626, 7744, 2400, 512, 73, 12, 0, 1, 1024, 19682, 65536, 78126, 46592, 16806, 4096, 703, 104, 10, 0

Offset: 1

Ilya Gutkovskiy, Nov 26 2018

			Square array begins:
  1,  1,   1,    1,     1,     1,  ...
  1,  2,   4,    8,    16,    32,  ...
  0,  2,   8,   26,    80,   242,  ...
  1,  4,  16,   64,   256,  1024,  ...
  2,  6,  26,  126,   626,  3126,  ...
  0,  4,  32,  208,  1280,  7744,  ...

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 antidiagonals)
Index entries for sequences mentioned by Glaisher

Columns k=0..12 give A002654, A050469, A050470, A050471, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

Cf. A109974, A279394, A279396, A285425, A322081, A322082, A322083.

Table[Function[k, SeriesCoefficient[Sum[j^k x^j/(1 + x^(2 j)), {j, 1, n}], {x, 0, n}]][i - n], {i, 0, 12}, {n, 1, i}] // Flatten

T(n,k)={sumdiv(n, d, if(d%2, (-1)^((d-1)/2)*(n/d)^k))}
for(n=1, 10, for(k=0, 8, print1(T(n, k), ", ")); print); \\ Andrew Howroyd, Nov 26 2018

G.f. of column k: Sum_{j>=1} j^k*x^j/(1 + x^(2*j)).

A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

Original entry on oeis.org

1, -4, -4, 32, -4, -104, 32, 192, -4, -292, -104, 480, 32, -680, 192, 832, -4, -1160, -292, 1440, -104, -1536, 480, 2112, 32, -2604, -680, 2624, 192, -3368, 832, 3840, -4, -3840, -1160, 4992, -292, -5480, 1440, 5440, -104, -6728, -1536, 7392, 480, -7592, 2112, 8832, 32, -9412, -2604

Offset: 0

Michael Somos, Jun 05 2006

sign

Number 8 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 - 4*q - 4*q^2 + 32*q^3 - 4*q^4 - 104*q^5 + 32*q^6 + 192*q^7 - 4*q^8 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 85, Eq. (32.7).

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.
Frazer Jarvis and Helena A. Verrill, Supercongruences for the Catalan-Larcombe-French numbers, Ramanujan J (22) (2010) 171.
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. A000141, A002173, A050470, A128692.

A := Basis( ModularForms( Gamma1(4), 3), 51); A[1] - 4*A[2]; /* Michael Somos, May 24 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ q^2]^3 (QPochhammer[ q] / QPochhammer[ q^4])^2)^2, {q, 0, n}]; (* Michael Somos, May 24 2015 *)
a[ n_] := If[ n < 1, Boole[n == 0], -4 DivisorSum[ n, #^2 KroneckerSymbol[ -4, #] &]]; (* Michael Somos, May 24 2015 *)

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

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x + A)^4 * eta(x^2 + A)^6 / eta(x^4 + A)^4, n))};

Expansion of eta(q)^4 * eta(q^2)^6 / eta(q^4)^4 in powers of q.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2 * (u - v)^2 - 4 * u*w * (v - w) * (u - 2*v).
Euler transform of period 4 sequence [ -4, -10, -4, -6, ...].
G.f.: 1 - 4 * Sum_{k>0} A056594(k-1) * k^2 * x^k / (1 - x^k).
Expansion of phi(-q)^2 * phi(-q^2)^4 in powers of q where phi(), psi() are Ramanujan theta functions. - Michael Somos, Aug 15 2007
G.f.: (Sum_{k in Z} (-1)^k * x^k^2)^2 * (Sum_{k in Z} (-1)^k * x^(2*k^2))^4.
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 128 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A050470.
a(n) = -4 * A002173(n) unless n=0.
Convolution of A000141 and A128692.

A122854 Expansion of phi(q)^2*psi(q)^4 in powers of q where phi(),psi() are Ramanujan theta functions.

Original entry on oeis.org

1, 8, 26, 48, 73, 120, 170, 208, 290, 360, 384, 528, 651, 656, 842, 960, 960, 1248, 1370, 1360, 1682, 1848, 1898, 2208, 2353, 2320, 2810, 3120, 2880, 3480, 3722, 3504, 4420, 4488, 4224, 5040, 5330, 5208, 5760, 6240, 5905, 6888, 7540, 6736, 7922, 8160, 7680

Offset: 0

Michael Somos, Sep 14 2006

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.

A050458(2n+1) = A050470(2n+1) = a(n).
Cf. A152584.
Cf. A000122, A000700, A010054, A121373.

a[n_]:= SeriesCoefficient[q^(-1/2)*(EllipticTheta[2, 0, q^(1/2)]^4 * EllipticTheta[3, 0, q]^2)/16, {q, 0, n}]; Table[A122854[n], {n, 0, 50}] (* G. C. Greubel, Jan 04 2018 *)

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

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^18/(eta(x+A)^8*eta(x^4+A)^4), n))}

Expansion of q^(-1/2)eta(q^2)^18/(eta(q)^8*eta(q^4)^4) in powers of q.
Euler transform of period 4 sequence [ 8, -10, 8, -6, ...].
a(n)=b(2n+1) where b(n) is multiplicative and b(2^e) = 0^e, b(p^e) = ((p^2)^(e+1)-1)/(p^2-1) if p == 1 (mod 4), b(p^e) = ((p^2)^(e+1)-(-1)^(e+1))/(p^2+1) if p == 3 (mod 4).
G.f.: Sum_{k>0 odd} k^2*x^k/(1+x^(2k)) = Product_{k>0} (1-x^(2k))^6*(1+x^k)^8/(1+x^(2k))^4.
Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^3/24 = 1.291928... (A152584). - Amiram Eldar, Dec 29 2023

A138501 Expansion of (eta(q)^2 * eta(q^4)^4 / eta(q^2)^3)^2 in powers of q.

Original entry on oeis.org

1, -4, 8, -16, 26, -32, 48, -64, 73, -104, 120, -128, 170, -192, 208, -256, 290, -292, 360, -416, 384, -480, 528, -512, 651, -680, 656, -768, 842, -832, 960, -1024, 960, -1160, 1248, -1168, 1370, -1440, 1360, -1664, 1682, -1536, 1848, -1920, 1898, -2112, 2208, -2048, 2353, -2604

Offset: 1

Michael Somos, Mar 20 2008

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

			G.f. = q - 4*q^2 + 8*q^3 - 16*q^4 + 26*q^5 - 32*q^6 + 48*q^7 - 64*q^8 + 73*q^9 + ...

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. A050470, A138502.

a[ n_] := If[ n < 1, 0, -(-1)^n DivisorSum[ n, #^2 Mod[n/#, 2] (-1)^Quotient[n/#, 2] &]]; (* Michael Somos, Aug 26 2015 *)
a[ n_] := SeriesCoefficient[ q (QPochhammer[ q]^2 QPochhammer[ q^4]^4 / QPochhammer[ q^2]^3)^2, {q, 0, n}]; (* Michael Somos, Aug 26 2015 *)

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

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

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

Expansion of q * (phi(-q) * psi(q^2)^2)^2 in powers of q where phi(), psi() are Ramanujan theta functions.
Euler transform of period 4 sequence [ -4, 2, -4, -6, ...].
a(n) is multiplicative with a(2^e) = -(4^e) if e>0, a(p^e) = ((p^2)^(e+1) - 1) / (p^2 - 1) if p == 1 (mod 4), a(p^e) = ((p^2)^(e+1) - (-1)^(e+1)) / (p^2 + 1) if p == 3 (mod 4).
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = 2 (t/i)^3 g(t) where q = exp(2 Pi i t) and g() is the g.f. for A138502.
G.f.: x * (Product_{k>0} (1 - x^k)^3 * (1 + x^k) * (1 + x^(2*k))^4)^2.
a(n) = -(-1)^n * A050470(n).

A364013 Expansion of Sum_{k>0} k^2 * x^k / (1 + x^(3*k)).

Original entry on oeis.org

1, 4, 9, 15, 25, 36, 50, 60, 81, 99, 121, 135, 170, 200, 225, 239, 289, 324, 362, 371, 450, 483, 529, 540, 626, 680, 729, 750, 841, 891, 962, 956, 1089, 1155, 1250, 1215, 1370, 1448, 1530, 1483, 1681, 1800, 1850, 1811, 2025, 2115, 2209, 2151, 2451, 2479, 2601, 2550, 2809, 2916, 3026

Offset: 1

Seiichi Manyama, Jul 01 2023

nonn

Cf. A364011, A364012.

Cf. A050470.

a[n_] := -DivisorSum[n, (-1)^(n/#) * #^2 &, Mod[n/#, 3] == 1 &]; Array[a, 100] (* Amiram Eldar, Jul 01 2023 *)

a(n) = -sumdiv(n, d, (n/d%3==1)*(-1)^(n/d)*d^2);

a(n) = -Sum_{d|n, n/d==1 (mod 3)} (-1)^(n/d) * d^2.

cp's OEIS Frontend

A321834 a(n) = Sum_{d|n, n/d==1 mod 4} d^10 - Sum_{d|n, n/d==3 mod 4} d^10.

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A321835 a(n) = Sum_{d|n, n/d==1 mod 4} d^11 - Sum_{d|n, n/d==3 mod 4} d^11.

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A050461 a(n) = Sum_{d|n, n/d=1 mod 4} d^2.

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A050465 a(n) = Sum_{d|n, n/d=3 mod 4} d^2.

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A322084 Square array A(n,k), n >= 1, k >= 0, read by antidiagonals: A(n,k) = Sum_{d|n, n/d==1 (mod 4)} d^k - Sum_{d|n, n/d==3 (mod 4)} d^k.

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A120030 Expansion of theta_4(q)^2*theta_4(q^2)^4 in powers of q.

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A122854 Expansion of phi(q)^2*psi(q)^4 in powers of q where phi(),psi() are Ramanujan theta functions.

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