A008457 -id:A008457 - cp's OEIS frontend

A000143 Number of ways of writing n as a sum of 8 squares.

1, 16, 112, 448, 1136, 2016, 3136, 5504, 9328, 12112, 14112, 21312, 31808, 35168, 38528, 56448, 74864, 78624, 84784, 109760, 143136, 154112, 149184, 194688, 261184, 252016, 246176, 327040, 390784, 390240, 395136, 476672, 599152, 596736, 550368, 693504, 859952

Offset: 0

N. J. A. Sloane

The relevant identity for the o.g.f. is theta_3(x)^8 = 1 + 16*Sum_{j >= 1} j^3*x^j/(1 - (-1)^j*x^j). See the Hardy-Wright reference, p. 315. - Wolfdieter Lang, Dec 08 2016

			1 + 16*q + 112*q^2 + 448*q^3 + 1136*q^4 + 2016*q^5 + 3136*q^6 + 5504*q^7 + ...

N. J. Fine, Basic Hypergeometric Series and Applications, Amer. Math. Soc., 1988; p. 77, Eq. (31.61); p. 79 Eq. (32.32).
E. Grosswald, Representations of Integers as Sums of Squares. Springer-Verlag, NY, 1985, p. 121.
G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers. 3rd ed., Oxford Univ. Press, 1954, pp. 314 - 315.

T. D. Noe, Table of n, a(n) for n = 0..10000
J. M. Borwein and K.-K. S. Choi, On Dirichlet series for sums of squares, Ramanujan J. 7 (1-3) (2003) 95-127.
H. H. Chan and C. Krattenthaler, Recent progress in the study of representations of integers as sums of squares, arXiv:math/0407061 [math.NT], 2004.
Shi-Chao Chen, Congruences for rs(n), Journal of Number Theory, Volume 130, Issue 9, September 2010, Pages 2028-2032.
F. Michel Dekking, Morphisms, Symbolic Sequences, and Their Standard Forms, Journal of Integer Sequences, Vol. 19 (2016), Article 16.1.1.
P. J. C. Lamont, The number of Cayley integers of given norm, Proceedings of the Edinburgh Mathematical Society, 25.1 (1982): 101-103. See (5).
S. C. Milne, Infinite families of exact sums of squares formulas, Jacobi elliptic functions, continued fractions and Schur functions, Ramanujan J., 6 (2002), 7-149.
M. Peters, Sums of nine squares, Acta Arith., 102 (2002), 131-135.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions
Index entries for sequences related to sums of squares

8th column of A286815. - Seiichi Manyama, May 27 2017
Row d=8 of A122141.
Cf. A008457, A035016, A010815.
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).
Cf. A004018, A000118, A000141 for the expansion of the powers of 2, 4, 6 of theta_3(x).

# JacobiTheta3 is defined in A000122.
A000143List(len) = JacobiTheta3(len, 8)
A000143List(37) |> println # Peter Luschny, Mar 12 2018

(sum(x^(m^2),m=-10..10))^8;
with(numtheory); rJ := n-> if n=0 then 1 else 16*add((-1)^(n+d)*d^3, d in divisors(n)); fi; [seq(rJ(n),n=0..100)]; # N. J. A. Sloane, Sep 15 2018

Table[SquaresR[8, n], {n, 0, 33}] (* Ray Chandler, Dec 06 2006 *)
SquaresR[8,Range[0,50]] (* Harvey P. Dale, Aug 26 2011 *)
QP = QPochhammer; s = (QP[q^2]^5/(QP[q]*QP[q^4])^2)^8 + O[q]^40; CoefficientList[s, q] (* Jean-François Alcover, Dec 01 2015, adapted from PARI *)

{a(n) = if( n<1, n==0, 16 * (-1)^n * sumdiv( n, d, (-1)^d * d^3))}

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x^2 + A)^5 / (eta(x + A) * eta(x^4 + A))^2)^8, n))} /* Michael Somos, Sep 25 2005 */

from math import prod
from sympy import factorint
def A000143(n): return prod((p**(3*(e+1))-(1 if p&1 else 15))//(p**3-1) for p, e in factorint(n).items())<<4 if n else 1 # Chai Wah Wu, Jun 21 2024

Q = DiagonalQuadraticForm(ZZ, [1]*8)
Q.representation_number_list(60) # Peter Luschny, Jun 20 2014

Expansion of theta_3(z)^8. Also a(n)=16*(-1)^n*Sum_{0

Expansion of phi(q)^8 in powers of q where phi() is a Ramanujan theta function. - Michael Somos, Mar 21 2008

Expansion of (eta(q^2)^5 / (eta(q) * eta(q^4))^2)^8 in powers of q. - Michael Somos, Sep 25 2005

G.f.: s(2)^40/(s(1)*s(4))^16, where s(k) := subs(q=q^k, eta(q)) and eta(q) is Dedekind's function, cf. A010815. [Fine]

Euler transform of period 4 sequence [16, -24, 16, -8, ...]. - Michael Somos, Apr 10 2005

a(n) = 16 * b(n) and b(n) is multiplicative with b(p^e) = (p^(3*e+3) - 1) / (p^3 - 1) -2[p<3]. - Michael Somos, Sep 25 2005

G.f.: 1 + 16 * Sum_{k>0} k^3 * x^k / (1 - (-x)^k). - Michael Somos, Sep 25 2005

A035016(n) = (-1)^n * a(n). 16 * A008457(n) = a(n) unless n=0.

Dirichlet g.f.: Sum_{n>=1} a(n)/n^s = 16*(1 - 2^(1-s) + 4^(2-s))*zeta(s)*zeta(s-3). [Borwein and Choi], R. J. Mathar, Jul 02 2012

a(n) = (16/n)*Sum_{k=1..n} A186690(k)*a(n-k), a(0) = 1. - Seiichi Manyama, May 27 2017

Sum_{k=1..n} a(k) ~ Pi^4 * n^4 /24. - Vaclav Kotesovec, Jul 12 2024

A064027 a(n) = (-1)^nSum_{d|n} (-1)^dd^2.

Original entry on oeis.org

1, 3, 10, 19, 26, 30, 50, 83, 91, 78, 122, 190, 170, 150, 260, 339, 290, 273, 362, 494, 500, 366, 530, 830, 651, 510, 820, 950, 842, 780, 962, 1363, 1220, 870, 1300, 1729, 1370, 1086, 1700, 2158, 1682, 1500, 1850, 2318, 2366, 1590, 2210, 3390, 2451, 1953

Offset: 1

Vladeta Jovovic, Sep 11 2001

			L.g.f.: L(x) = x + 3*x^2/2 + 10*x^3/3 + 19*x^4/4 + 26*x^5/5 + 30*x^6/6 + ...
where exp(L(x)) = 1 + x + 2*x^2 + 5*x^3 + 10*x^4 + 18*x^5 + 32*x^6 + 59*x^7 + 106*x^8 + 181*x^9 + ... + A224364(n)*x^n + ... - _Paul D. Hanna_, Apr 04 2013

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Paul D. Hanna, Table of n, a(n) for n = 1..1000
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.

Cf. A001157, A002129, A008457, A050999, A224364.

Cf. A321543 - A321565, A321807 - A321836 for related sequences.

m:=60; R:=PowerSeriesRing(Integers(), m); Coefficients(R!( (&+[k^2*x^k/(1-(-x)^k): k in [1..m]]) )); // G. C. Greubel, Nov 09 2018

a[n_] := (-1)^n DivisorSum[n, (-1)^# * #^2 &]; Array[a, 50] (* Jean-François Alcover, Dec 23 2015 *)
a[n_] := If[OddQ[n], 1, (1 - 6/(4^(IntegerExponent[n, 2] + 1) - 1))] * DivisorSigma[2, n]; Array[a, 100] (* Amiram Eldar, Sep 21 2023 *)

{a(n)=if(n<1, 0, (-1)^n*sumdiv(n^1, d, (-1)^d*d^2))} \\ Paul D. Hanna, Apr 04 2013

Multiplicative with a(2^e) = (4^(e+1)-7)/3, a(p^e) = (p^(2*e+2)-1)/(p^2-1), p > 2.
a(n) = (-1)^n*(A001157(n) - 2*A050999(n)).
Logarithmic derivative of A224364. - Paul D. Hanna, Apr 04 2013
Bisection: a(2*k-1) = A001157(2*k-1), a(2*k) = 4*A001157(k) - A050999(2*k), k >= 1. In the Hardy reference a(n) = sigma^*2(n). - _Wolfdieter Lang, Jan 07 2017
G.f.: Sum_{k>=1} k^2*x^k/(1 - (-x)^k). - Ilya Gutkovskiy, Nov 09 2018
Sum_{k=1..n} a(k) ~ 7 * zeta(3) * n^3 / 24. - Vaclav Kotesovec, Nov 10 2018
Dirichlet g.f.: zeta(s) * zeta(s-2) * (1 - 1/2^(s-1) + 1/2^(2*s-3)). - Amiram Eldar, Sep 21 2023

A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

Original entry on oeis.org

1, -7, 28, -71, 126, -196, 344, -583, 757, -882, 1332, -1988, 2198, -2408, 3528, -4679, 4914, -5299, 6860, -8946, 9632, -9324, 12168, -16324, 15751, -15386, 20440, -24424, 24390, -24696, 29792, -37447, 37296, -34398, 43344, -53747, 50654, -48020, 61544, -73458

Offset: 1

Michael Somos, Mar 21 2008

Also, expansion of (1 - phi(-q)^8) / 16 in powers of q where phi() is a Ramanujan theta function.

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

			G.f. = q - 7*q^2 + 28*q^3 - 71*q^4 + 126*q^5 - 196*q^6 + 344*q^7 - 583*q^8 + ...

G. C. Greubel, 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).
Heekyoung Hahn, Convolution sums of some functions on divisors, arXiv:1507.04426 [math.NT], 2015.
Michael Somos, Introduction to Ramanujan theta functions.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions.
Index entries for sequences mentioned by Glaisher.

Divisor sums Sum_{d|n} (-1)^(d-1)*d^k: A048272 (k = 0), A002129 (k = 1), A321543 (k = 2), A279395 (k = 4, unsigned), A321544 - A321551 (k = 5 to k = 12).

Cf. A008457, A035016.

with(numtheory):
a := n -> add( (-1)^(d-1)*d^3, d in divisors(n) ): seq(a(n), n = 1..40);
#  Peter Bala, Jan 11 2021

a[ n_] := If[ n < 0, 0, DivisorSum[ n, -(-1)^# #^3&]]; (* Michael Somos, Sep 25 2015 *)
a[ n_] := SeriesCoefficient[ (1 - EllipticTheta[ 4, 0, q]^8) / 16, {q, 0, n}]; (* Michael Somos, Sep 25 2015 *)
nmax = 40; Rest[CoefficientList[Series[-Product[((1-q^k)/(1+q^k))^8, {k, 1, nmax}]/16, {q, 0, nmax}], q]] (* Vaclav Kotesovec, Sep 26 2015 *)
f[p_, e_] := (p^(3*e + 3) - 1)/(p^3 - 1); f[2, e_] := 2 - (2^(3*e + 3) - 1)/7; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 40] (* Amiram Eldar, Nov 04 2022 *)

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

Expansion of (1 - (eta(q)^2 / eta(q^2))^8) / 16 in powers of q.
a(n) is multiplicative with a(2^e) = -(8^(e+1) - 15) / 7, a(p^e) = ((p^3)^(e+1) - 1) / (p^3 - 1).
G.f.: Sum_{k>0} k^3 * -(-x)^k / (1 - x^k).
a(n) = -(-1)^n * A008457(n). -16 * a(n) = A035016(n) unless n=0.
G.f.: Sum_{n >= 1} x^n*(1 - 4*x^n + x^(2*n))/(1 + x^n)^4. - Peter Bala, Jan 11 2021

Simpler definition from N. J. A. Sloane, Nov 23 2018

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.

Original entry on oeis.org

1, 1, 0, 1, 1, 2, 1, 3, 4, 1, 1, 7, 10, 5, 2, 1, 15, 28, 19, 6, 0, 1, 31, 82, 71, 26, 4, 2, 1, 63, 244, 271, 126, 30, 8, 2, 1, 127, 730, 1055, 626, 196, 50, 13, 3, 1, 255, 2188, 4159, 3126, 1230, 344, 83, 13, 0, 1, 511, 6562, 16511, 15626, 7564, 2402, 583, 91, 6, 2, 1, 1023, 19684, 65791, 78126, 45990, 16808, 4367, 757, 78, 12, 2

Offset: 1

Wolfdieter Lang, Jan 10 2017

The array A(k, n) = sigma^*A279395)%20=%20Sum">(k)(n) (notation of the Hardy reference, given also in a comment in A279395) = Sum{ d >= 1, d divides n} (-1)^(n-d)*d^k, for k >= 0 and n >=1, has the rows A112329, A113184, A064027, A008457, A279395, for k=0..4.
The triangle T(n, m) is obtained from the array A(k, n) read by upwards antidiagonals, with offset n=1.
The diagonals of triangle T are the rows of the array A. Each diagonal is multiplicative. See the given A-numbers above.
The row sums are given in A279397.
The column sums (with offset 0) are A000012, A000225, A034472, A099393, A034474, .. with o.g.f. G(m, z) = (-1)^m*Sum_{d  | m} (-1)^d/(1 - d*z), m >= 1.

			The triangle T(n, m) begins:
n\m 1   2    3    4    5    6   7  8  9 10
1:  1
2:  1   0
3:  1   1    2
4:  1   3    4    1
5:  1   7   10    5    2
6:  1  15   28   19    6    0
7:  1  31   82   71   26    4   2
8:  1  63  244  271  126   30   8  2
9:  1 127  730 1055  626  196  50 13  3
10: 1 255 2188 4159 3126 1230 344 83 13  0
...
n = 11: 1 511 6562 16511 15626 7564 2402 583 91 6 2,
n = 12: 1 1023 19684 65791 78126 45990 16808 4367 757 78 12 2.
n = 13: 1 2047 59050 262655 390626 277876 117650 33823 6643 882 122 20 2,
n = 14: 1 4095 177148 1049599 1953126 1673310 823544 266303 59293 9390 1332 190 14 0,
n = 15: 1 8191 531442 4196351 9765626 10058524 5764802 2113663 532171 96906 14642 1988 170 8 4.
...

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Cf. A279394, A112329, A113184, A064027, A008457, A279395, A000012, A000225, A034472, A099393, A034474.

T(n, m) = Sum_{ d >= 1, d divides m} (-1)^(m-d)*d^(n-m) = sigma^*_(n-m)(m), n >= 1, m = 1,2, ..., n. For the definition of
sigma^*_(k)(n) see the Hardy reference or a comment in A279395.
O.g.f triangle T: G(z, x) = Sum_{m>=0}
G(m, z)*(x*z)^m, with the column o.g.f. G( m, z) (with offset 0) given in a comment above.

A279395 a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.

Original entry on oeis.org

1, 15, 82, 271, 626, 1230, 2402, 4367, 6643, 9390, 14642, 22222, 28562, 36030, 51332, 69903, 83522, 99645, 130322, 169646, 196964, 219630, 279842, 358094, 391251, 428430, 538084, 650942, 707282, 769980, 923522, 1118479, 1200644, 1252830, 1503652, 1800253, 1874162, 1954830, 2342084, 2733742

Offset: 1

Wolfdieter Lang, Jan 09 2017

This is the k=4 member of the family sigma^*_k(n), defined in the Hardy reference, which is sigma_k(2*j+1) if n = 2*j+1 and sigma_k^e(2*j) - sigma_k^o(2*j) if n=2*j, where the superscript e and o stands for a restriction to even and odd divisors in the sum of their k-th powers, respectively.

G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, p. 142.

Amiram Eldar, 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. A112329 (k=0), A113184 (k=1), A064027 (k=2), A008457(k=3).

[&+[(-1)^(n-d)*d^4:d in Divisors(n)]:n in [1..40]]; // Marius A. Burtea, Aug 17 2019

# A version with signs - N. J. A. Sloane, Nov 23 2018
zet1:=(n,i)->add((-1)^(d-1)*d^i, d in divisors(n));
szet1:=i->[seq(zet1(n,i),n=1..120)];
szet1(4);

f[p_, e_] := If[p == 2, (2^(4*(e + 1)) - 31)/15, (p^(4*(e + 1)) - 1)/(p^4 - 1)]; a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 40] (* Amiram Eldar, Aug 17 2019 *)

a(n) = sumdiv(n, d, (-1)^(n-d)*d^4); \\ Michel Marcus, Jan 09 2017

a(n) = Sum_{ d >= 1, d divides n} (-1)^(n-d)*d^4.
Bisection: a(2*j-1) = A001159(2*j-1), a(2*j) = 16*A001159(j) - A051001(j), j >= 1. See the comment above for k=4, and the Hardy reference.
G.f.: Sum_{k>=1} k^4*x^k/(1-(-x)^k).
Multiplicative with a(2^k) = 2^4*(2^(4*k)-1)/(2^4-1) - 1 = (2^(4*(k+1)) - 31)/15 and a(p^k) = (p^(4*(k+1))-1)/(p^4-1) for primes p > 2 (see A001159).

A092820 a(0) = 1; for n>0, a(n) = 16 times sum of cubes of divisors of n.

Original entry on oeis.org

1, 16, 144, 448, 1168, 2016, 4032, 5504, 9360, 12112, 18144, 21312, 32704, 35168, 49536, 56448, 74896, 78624, 109008, 109760, 147168, 154112, 191808, 194688, 262080, 252016, 316512, 327040, 401792, 390240, 508032, 476672, 599184, 596736, 707616, 693504

Offset: 0

N. J. A. Sloane, Sep 11 2004

nonn

Cf. A004009, A008457, A035016.

A302857 Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^32 - 1) / 64.

Original entry on oeis.org

1, 31, 620, 8991, 100750, 908052, 6767672, 42576671, 229829413, 1079371826, 4466377700, 16483884820, 54927684614, 167288627912, 471202341864, 1240559020831, 3079711709682, 7258966849915, 16333838548700, 35251970265650, 73278213928864, 147241116313756, 286838266666792

Offset: 1

Seiichi Manyama, Apr 14 2018

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Expansion of ((1 + 2 * Sum_{k>=1} q^(k^2))^(2^m) - 1) / 2^(m + 1): A002654 (m=1), A046897 (m=2), A008457 (m=3), A302855 (m=4), this sequence (m=5).

Cf. A302856.

a(n) = A302856(n) / 64.

A177155 G.f.: exp( Integral (theta_3(x)^8-1)/(16x) dx ), where theta_3(x) = 1 + Sum_{n>=1} 2*x^(n^2) is a Jacobi theta function.

Original entry on oeis.org

1, 1, 4, 13, 35, 87, 217, 539, 1291, 2999, 6880, 15595, 34738, 76202, 165282, 354655, 752546, 1580514, 3289337, 6787085, 13887937, 28195434, 56824772, 113729640, 226104615, 446665922, 877063515, 1712252521, 3324245063, 6419561961

Offset: 0

Paul D. Hanna, May 03 2010, May 08 2010

nonn

Compare to g.f. of partitions in which no parts are multiples of 4:

g.f. of A001935 = exp( Integral (theta_3(x)^4-1)/(8x) dx ).

			G.f.: A(x) = 1 + x + 4*x^2 + 13*x^3 + 35*x^4 + 87*x^5 +...
log(A(x)) = x + 7*x^2/2 + 28*x^3/3 + 71*x^4/4 + 126*x^5/5 +...+ A008457(n)*x^n/n +...

Seiichi Manyama, Table of n, a(n) for n = 0..10000

Cf. A138503, A008457, A001935, A000143, A307462.

nmax = 40; Abs[CoefficientList[Series[Product[1/(1 - x^k)^((-1)^k*k^2), {k, 1, nmax}], {x, 0, nmax}], x]] (* Vaclav Kotesovec, Apr 10 2019 *)
nmax = 40; CoefficientList[Series[Product[(1 + x^(2*k - 1))^((2*k - 1)^2)/(1 - x^(2*k))^(4*k^2), {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Apr 10 2019 *)

{a(n)=polcoeff(exp(sum(m=1,n, sumdiv(m,d,(-1)^(m-d)*d^3)*x^m/m)+x*O(x^n)),n)}

{a(n)=local(theta3=1+sum(m=1,sqrtint(2*n+2),2*x^(m^2)+x*O(x^n)));polcoeff(exp(intformal((theta3^8-1)/(16*x))),n)}

G.f.: exp( Sum_{n>=1} A008457(n)*x^n/n ) where A008457(n) = Sum_{d|n} (-1)^(n-d)*d^3.

a(n) ~ exp(2*Pi*n^(3/4)/3 - Zeta(3)/Pi^2) / (4*n^(5/8)). - Vaclav Kotesovec, Apr 10 2019

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cp's OEIS Frontend

A190623 Mobius transform of A008457.

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A122141 Array: T(d,n) = number of ways of writing n as a sum of d squares, read by ascending antidiagonals.

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A000143 Number of ways of writing n as a sum of 8 squares.

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A064027 a(n) = (-1)^n*Sum_{d|n} (-1)^d*d^2.

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A138503 a(n) = Sum_{d|n} (-1)^(d-1)*d^3.

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A279396 Triangle read by rows T(n, m) = sigma^*(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^*(k)(n) given in a comment in A279395.

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A064027 a(n) = (-1)^nSum_{d|n} (-1)^dd^2.

A279396 Triangle read by rows T(n, m) = sigma^(n-m)(m), n >= 1, m = 1, 2, ..., n, with sigma^(k)(n) given in a comment in A279395.