A101455 -id:A101455 - cp's OEIS frontend

A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

1, 16, 80, 256, 626, 1280, 2400, 4096, 6481, 10016, 14640, 20480, 28562, 38400, 50080, 65536, 83522, 103696, 130320, 160256, 192000, 234240, 279840, 327680, 391251, 456992, 524960, 614400, 707282, 801280, 923520, 1048576, 1171200

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Dirichlet convolution of A000583 = n^4 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005
Called E'_4(n) by Hardy.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = x + 16*x^2 + 80*x^3 + 256*x^4 + 626*x^5 + 1280*x^6 + 2400*x^7 + 4096*x^8 + ...

Emil Grosswald, Representations of Integers as Sums of Squares, Springer-Verlag, NY, 1985, p. 120.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, Chelsea Publishing Company, New York 1959, p. 135, section 9.3. MR0106147 (21 #4881)

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).
Michael Somos, Introduction to Ramanujan theta functions, 2019.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Cf. A000122, A000583, A000700, A010054, A101455, A121373, A175571, A204342, A204372.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, this sequence, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

A := Basis( ModularForms( Gamma1(4), 5), 34); A[2] + 16*A[3]; /* Michael Somos, May 03 2015 */

edashed[r_,n_] := Plus@@(Select[Divisors[n], Mod[n/#,4] == 1 &]^r) - Plus@@(Select[Divisors[n], Mod[n/#,4] == 3 &]^r); edashed[4,#] &/@Range[33] (* Ant King, Nov 10 2012 *)
a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x^2] (EllipticTheta[ 2, 0, x]^8 + 4 EllipticTheta[ 2, 0, x^2]^8) / 256, {x, 0, 2 n}]; (* Michael Somos, Jan 11 2015 *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(4*e+4) - s[p]^(e+1))/(p^4 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

{a(n) = if( n<1, 0, sumdiv( n, d, (n/d%2) * (-1)^((n/d - 1)/2) * d^4))}; /* Michael Somos, Sep 12 2005 */

{a(n) = if( n<1, 0, sumdiv( n, d, d^4 * kronecker( -4, n\d)))}; /* Michael Somos, Jan 14 2012 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( eta(x^2 + A)^2 * eta(x^4 + A)^4 * (eta(x + A)^4 + 20 * x * eta(x^4 + A)^8 / eta(x + A)^4), n))}; /* Michael Somos, Jan 14 2012 */

a(2*n + 1) = A204342(n). a(2*n) = 16 * a(n).
G.f.: Sum_{n>=1} n^4*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Michael Somos, Jan 14 2012: (Start)
Expansion of eta(q^2)^2 * eta(q^4)^4 * (eta(q)^4 + 20 * eta(q^4)^8 / eta(q)^4) in powers of q.
a(n) is multiplicative with a(2^e) = 16^e, a(p^e) = ((p^4)^(e+1) - 1) / (p^4 - 1) if p == 1 (mod 4), a(p^e) = ((p^4)^(e+1) - (-1)^(e+1)) / (p^4 + 1) if p == 3 (mod 4). (End)
From Michael Somos, Jan 15 2012: (Start)
Expansion of theta_3(q^2) * (theta_2(q)^8 + 4 * theta_2(q^2)^8) / 256 in powers of q^2.
Expansion of x * phi(x)^2 * (psi(x)^8 + 4 * x * psi(x^2)^8) in powers of x where phi(), psi() are Ramanujan theta functions. (End)
G.f. is a period 1 Fourier series which satisfies f(-1 / (4 t)) = (1/2) (t/i)^5 g(t) where q = exp(2 Pi i t) and g(t) is the g.f. for A204372. - Michael Somos, May 03 2015
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(4*e+4) - A101455(p)^(e+1))/(p^4 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^5 / 5, where c = 5*Pi^5/1536 (A175571). (End)
a(n) = Sum_{d|n} (n/d)^4*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.

Original entry on oeis.org

0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0, 1, 0, 1, 0, -1, 0, -1, 0

Offset: 0

Michael Somos, Apr 10 2011

			G.f. = x + x^3 - x^5 - x^7 + x^9 + x^11 - x^13 - x^15 + x^17 + x^19 - x^21 + ...

G. C. Greubel, Table of n, a(n) for n = 0..2500
Michael Somos, Rational Function Multiplicative Coefficients
Eric Weisstein's World of Mathematics, Kronecker Symbol.
Wikipedia, Kronecker Symbol.
Index entries for linear recurrences with constant coefficients, signature (0,0,0,-1).

Cf. A002325, A057077, A091337, A101455, A257170.

m:=60; R:=PowerSeriesRing(Integers(), m); [0] cat Coefficients(R!(x*(1+x^2)/(1+x^4))); // G. C. Greubel, Aug 02 2018

Table[KroneckerSymbol[-2, n], {n, 0, 104}] (* Wolfdieter Lang, May 30 2013 *)
a[ n_] := Mod[n, 2] (-1)^Quotient[ n, 4]; (* Michael Somos, Apr 17 2015 *)
CoefficientList[Series[x*(1+x^2)/(1+x^4), {x, 0, 60}], x] (* G. C. Greubel, Aug 02 2018 *)
LinearRecurrence[{0,0,0,-1},{0,1,0,1},120] (* or *) PadRight[{},120,{0,1,0,1,0,-1,0,-1}] (* Harvey P. Dale, Jan 25 2023 *)

PARI
```
{a(n) = (n%2) * (-1)^(n\4)};
```

x='x+O('x^60); concat([0], Vec(x*(1+x^2)/(1+x^4))) \\ G. C. Greubel, Aug 02 2018

Euler transform of length 8 sequence [0, 1, 0, -2, 0, 0, 0, 1].
a(n) is multiplicative with a(2^e) = 0^e, a(p^e) = 1 if p == 1 or 3 (mod 8), a(p^e) = (-1)^e if p == 5 or 7 (mod 8).
G.f.: x * (1 - x^4)^2/((1 - x^2)*(1 - x^8)) = (x + x^3)/(1 + x^4).
a(-n) = -a(n) = a(n+4).
a(n+2) = A091337(n).
a(2*n) = 0, a(2*n+1) = A057077(n).
G.f.: x/(1 - x^2/(1 + 2*x^2/(1 - x^2))). - Michael Somos, Jan 03 2013
a(n) = ((-2)/n), where (k/n) is the Kronecker symbol. Period 8. See the Eric Weisstein link. - Wolfdieter Lang, May 29 2013
a(n) = A257170(n) unless n = 0.
From Jianing Song, Nov 14 2018: (Start)
a(n) = sqrt(2)*sin(Pi*n/2)*cos(Pi*n/4).
E.g.f.: sqrt(2)*sin(x/sqrt(2))*cosh(x/sqrt(2)).
Moebius transform of A002325.
a(n) = A091337(n)*A101455(n).
a(n) = ((-2)^(2*i+1)/n) for all integers i >= 0, where (k/n) is the Kronecker symbol. (End)
a(n) = A014017(n-1)+A014017(n-3). - R. J. Mathar, Dec 17 2024

A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

Original entry on oeis.org

1, 8, 26, 64, 126, 208, 342, 512, 703, 1008, 1330, 1664, 2198, 2736, 3276, 4096, 4914, 5624, 6858, 8064, 8892, 10640, 12166, 13312, 15751, 17584, 18980, 21888, 24390, 26208, 29790, 32768, 34580, 39312, 43092, 44992, 50654, 54864, 57148

Offset: 1

N. J. A. Sloane, Dec 23 1999

Multiplicative because it is the Dirichlet convolution of A000578 = n^3 and A101455 = [1 0 -1 0 1 0 -1 ...], which are both multiplicative. - Christian G. Bower, May 17 2005

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. A000578, A101455, A175572.

Glaisher's E'_i (i=0..12): A002654, A050469, A050470, this sequence, A050468, A321829, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

max = 40; s = Sum[n^3*x^(n-1)/(1+x^(2*n)), {n, 1, max}] + O[x]^max; CoefficientList[s, x] (* Jean-François Alcover, Dec 02 2015, after Vladeta Jovovic *)
s[n_] := If[OddQ[n], (-1)^((n-1)/2), 0]; (* A101455 *)
f[p_, e_] := (p^(3*e+3) - s[p]^(e+1))/(p^3 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

a(n) = sumdiv(n, d, d^3*(((n/d) % 4)==1)) - sumdiv(n, d, d^3*(((n/d) % 4)==3)); \\ Michel Marcus, Feb 16 2015

G.f.: Sum_{n>=1} n^3*x^n/(1+x^(2*n)). - Vladeta Jovovic, Oct 16 2002
From Amiram Eldar, Nov 04 2023: (Start)
Multiplicative with a(p^e) = (p^(3*e+3) - A101455(p)^(e+1))/(p^3 - A101455(p)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = A175572. (End)
a(n) = Sum_{d|n} (n/d)^3*sin(d*Pi/2). - Ridouane Oudra, Sep 26 2024

Offset changed from 0 to 1 by R. J. Mathar, Jul 15 2010

A084099 Expansion of (1+x)^2/(1+x^2).

Original entry on oeis.org

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

Offset: 0

Paul Barry, May 15 2003

Inverse binomial transform of A077860. Partial sums of A084100.
Transform of sqrt(1+2x)/sqrt(1-2x) (A063886) under the Chebyshev transformation A(x)->((1-x^2)/(1+x^2))*A(x/(1+x^2)). - Paul Barry, Oct 12 2004
Euler transform of length 4 sequence [2, -3, 0, 1]. - Michael Somos, Aug 04 2009

			G.f. = 1 + 2*x - 2*x^3 + 2*x^5 - 2*x^7 + 2*x^9 - 2*x^11 + 2*x^13 - 2*x^15 + ...

Colin Barker, Table of n, a(n) for n = 0..1000
Bérénice Delcroix-Oger and Clément Dupont, Lie-operads and operadic modules from poset cohomology, arXiv:2505.06094 [math.CO], 2025. See p. 28.
Index entries for linear recurrences with constant coefficients, signature (0,-1).

Cf. A063886, A077860, A084100, A101455, A180969.

[1] cat [Integers()!((1-(-1)^n)*(-1)^(n*(n-1)/2)): n in [1..100]]; // Wesley Ivan Hurt, Oct 27 2015

A084099:=n->(1-(-1)^n)*(-1)^((2*n-1+(-1)^n)/4): 1,seq(A084099(n), n=1..100); # Wesley Ivan Hurt, Oct 27 2015

CoefficientList[Series[(1+x)^2/(1+x^2),{x,0,110}],x] (* or *) Join[ {1}, PadRight[{},120,{2,0,-2,0}]] (* Harvey P. Dale, Nov 23 2011 *)

{a(n) = if( n<1, n==0, 2 * if( n%2, (-1)^(n\2)) )}; /* Michael Somos, Aug 04 2009 */

a(n) = if(n==0, 1, I*((-I)^n-I^n)) \\ Colin Barker, Oct 27 2015

Vec((1+x)^2/(1+x^2) + O(x^100)) \\ Colin Barker, Oct 27 2015

G.f.: (1+x)^2/(1+x^2).
a(n) = 2 * A101455(n) for n>0. - N. J. A. Sloane, Jun 01 2010
a(n+2) = (-1)^A180969(1,n)*((-1)^n - 1). - Adriano Caroli, Nov 18 2010
G.f.: 4*x + 2/(1+x)/G(0), where G(k) = 1 + 1/(1 - x*(2*k-1)/(x*(2*k+1) - 1/G(k+1))); (continued fraction). - Sergei N. Gladkovskii, Jun 19 2013
From Wesley Ivan Hurt, Oct 27 2015: (Start)
a(n) = (1-sign(n)*(-1)^n)*(-1)^floor(n/2).
a(n) = 2*(n mod 2)*(-1)^floor(n/2) for n>0, a(0)=1.
a(n) = (1-(-1)^n)*(-1)^(n*(n-1)/2) for n>0, a(0)=1. (End)
From Colin Barker, Oct 27 2015: (Start)
a(n) = -a(n-2).
a(n) = i*((-i)^n-i^n) for n>0, where i = sqrt(-1).
(End)

A175571 Decimal expansion of the Dirichlet beta function of 5.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

The value of the Dirichlet L-series L(m=4,r=2,s=4), see arXiv:1008.2547.

			0.99615782807708806400631936...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(5) ; x := evalf(x) ;

RealDigits[ DirichletBeta[5], 10, 105] // First (* Jean-François Alcover, Feb 20 2013, updated Mar 14 2018 *)

5*Pi^5/1536 \\ Charles R Greathouse IV, Jan 31 2018

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(5) \\ Charles R Greathouse IV, Jan 31 2018

Equals 5*Pi^5/1536 = Sum_{n>=1} A101455(n)/n^5, where Pi^5 = A092731. [corrected by R. J. Mathar, Feb 01 2018]
Equals Sum_{n>=0} (-1)^n/(2*n+1)^5. - Jean-François Alcover, Mar 29 2013
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^5)^(-1). - Amiram Eldar, Nov 06 2023

A175572 Decimal expansion of the Dirichlet beta function of 4.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

This is the value of the Dirichlet L-series for A101455 at s=4, see arXiv:1008.2547, L(m=4,r=2,s=4).

			0.988944551741105336108422633...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. (308).

G. C. Greubel, Table of n, a(n) for n = 0..10000
Mark W. Coffey, Summatory relations and prime products for the Stieltjes constants and other related results, arXiv:1701.07064, proposition 5.
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175571 (beta(5)), A175570 (beta(6)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(4) ; x := evalf(x) ;

RealDigits[ DirichletBeta[4], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(4) \\ Charles R Greathouse IV, Jan 31 2018

Equals Sum_{n>=1} A101455(n)/n^4. [corrected by R. J. Mathar, Feb 01 2018]
Equals (PolyGamma(3, 1/4) - PolyGamma(3, 3/4))/1536. - Jean-François Alcover, Jun 11 2015
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^4)^(-1). - Amiram Eldar, Nov 06 2023

A175570 Decimal expansion of the Dirichlet beta function of 6.

Original entry on oeis.org

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

Offset: 0

R. J. Mathar, Jul 15 2010

			0.998685222218438135441600...

L. B. W. Jolley, Summation of Series, Dover, 1961, eq. 308.

G. C. Greubel, Table of n, a(n) for n = 0..10000
Richard J. Mathar, Table of Dirichlet L-Series and Prime Zeta Modulo Functions for Small Moduli, arXiv:1008.2547 [math.NT], 2010-2015.
Eric Weisstein's World of Mathematics, Dirichlet Beta Function.
Wikipedia, Dirichlet beta function.

Cf. A003881 (beta(1)=Pi/4), A006752 (beta(2)=Catalan), A153071 (beta(3)), A175572 (beta(4)), A175571 (beta(5)), A258814 (beta(7)), A258815 (beta(8)), A258816 (beta(9)).

Cf. A101455.

DirichletBeta := proc(s) 4^(-s)*(Zeta(0,s,1/4)-Zeta(0,s,3/4)) ; end proc: x := DirichletBeta(6) ; x := evalf(x) ;

RealDigits[ DirichletBeta[6], 10, 105] // First (* Jean-François Alcover, Feb 11 2013, updated Mar 14 2018 *)

beta(x)=(zetahurwitz(x,1/4)-zetahurwitz(x,3/4))/4^x
beta(6) \\ Charles R Greathouse IV, Jan 31 2018

sumpos(n=1,(12288*n^5 - 30720*n^4 + 33280*n^3 - 19200*n^2 + 5808*n - 728)/(16777216*n^12 - 100663296*n^11 + 270532608*n^10 - 429916160*n^9 + 449249280*n^8 - 324796416*n^7 + 166445056*n^6 - 60899328*n^5 + 15793920*n^4 - 2833920*n^3 + 334368*n^2 - 23328*n + 729),1) \\ Charles R Greathouse IV, Feb 01 2018

Equals Sum_{n>=1} A101455(n)/n^6. [see arxiv:1008.2547, L(m=4,r=2,s=6)] [corrected by R. J. Mathar, Feb 01 2018]
Equals (PolyGamma(5, 1/4) - PolyGamma(5, 3/4))/491520. - Jean-François Alcover, Jun 11 2015
Equals Product_{p prime >= 3} (1 - (-1)^((p-1)/2)/p^6)^(-1). - Amiram Eldar, Nov 06 2023

A321829 a(n) = Sum_{d|n, n/d==1 mod 4} d^5 - Sum_{d|n, n/d==3 mod 4} d^5.

Original entry on oeis.org

1, 32, 242, 1024, 3126, 7744, 16806, 32768, 58807, 100032, 161050, 247808, 371294, 537792, 756492, 1048576, 1419858, 1881824, 2476098, 3201024, 4067052, 5153600, 6436342, 7929856, 9768751, 11881408, 14290100, 17209344, 20511150, 24207744, 28629150, 33554432, 38974100, 45435456, 52535556

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, A175570.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, this sequence, A321830, A321831, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, # ^5 &, 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^(5*e+5) - s[p]^(e+1))/(p^5 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321829(n)=factorback(apply(f->f[1]^(5*f[2]+5)\/(f[1]^5+f[1]%4-2),Col(factor(n)))), [1..40]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^5*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(5e+5)/(p^5 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^6 / 6, where c = A175570. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^5*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321830 a(n) = Sum_{d|n, n/d==1 mod 4} d^6 - Sum_{d|n, n/d==3 mod 4} d^6.

Original entry on oeis.org

1, 64, 728, 4096, 15626, 46592, 117648, 262144, 530713, 1000064, 1771560, 2981888, 4826810, 7529472, 11375728, 16777216, 24137570, 33965632, 47045880, 64004096, 85647744, 113379840, 148035888, 190840832, 244156251, 308915840, 386889776

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, A258814.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, this sequence, A321831, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^6 &, 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^(6*e+6) - s[p]^(e+1))/(p^6 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321830(n)=factorback(apply(f->f[1]^(6*f[2]+6)\/(f[1]^6+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^6*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(6e+6)/(p^6 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^7 / 7, where c = 61*Pi^7/184320 (A258814). - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^6*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

A321831 a(n) = Sum_{d|n, n/d==1 mod 4} d^7 - Sum_{d|n, n/d==3 mod 4} d^7.

Original entry on oeis.org

1, 128, 2186, 16384, 78126, 279808, 823542, 2097152, 4780783, 10000128, 19487170, 35815424, 62748518, 105413376, 170783436, 268435456, 410338674, 611940224, 893871738, 1280016384, 1800262812, 2494357760, 3404825446, 4584374272, 6103593751

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, A258815.
Cf. A321543 - A321565, A321807 - A321836 for similar sequences.
Glaisher's E'_i (i=0..12): A002654, A050469, A050470, A050471, A050468, A321829, A321830, this sequence, A321832, A321833, A321834, A321835, A321836.

s[n_,r_] := DivisorSum[n, #^7 &, 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^(7*e+7) - s[p]^(e+1))/(p^7 - s[p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 04 2023 *)

apply( A321831(n)=factorback(apply(f->f[1]^(7*f[2]+7)\/(f[1]^7+f[1]%4-2),Col(factor(n)))), [1..30]) \\ M. F. Hasler, Nov 26 2018

G.f.: Sum_{k>=1} k^7*x^k/(1 + x^(2*k)). - Ilya Gutkovskiy, Nov 26 2018
Multiplicative with a(p^e) = round(p^(7e+7)/(p^7 + p%4 - 2)), where p%4 is the remainder of p modulo 4. (Following R. Israel in A321833.) - M. F. Hasler, Nov 26 2018
Sum_{k=1..n} a(k) ~ c * n^8 / 8, where c = A258815. - Amiram Eldar, Nov 04 2023
a(n) = Sum_{d|n} (n/d)^7*sin(d*Pi/2). - Ridouane Oudra, Sep 27 2024

cp's OEIS Frontend

A050468 a(n) = Sum_{d|n, n/d=1 mod 4} d^4 - Sum_{d|n, n/d=3 mod 4} d^4.

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A188510 Expansion of x*(1 + x^2)/(1 + x^4) in powers of x.

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A050471 a(n) = Sum_{d|n, n/d=1 mod 4} d^3 - Sum_{d|n, n/d=3 mod 4} d^3.

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A084099 Expansion of (1+x)^2/(1+x^2).

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A175571 Decimal expansion of the Dirichlet beta function of 5.

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A175572 Decimal expansion of the Dirichlet beta function of 4.

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