A013973 -id:A013973 - cp's OEIS frontend

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

1, -42, -11088, -3774624, -1472710974, -617481728640, -270883381218912, -122585272771463040, -56747118995519331456, -26727350506044696990762, -12760853360973370821796320, -6159994719956314185540737376, -3000691311646502407278581263104, -1472883416501251994527873967792256

Offset: 0

N. J. A. Sloane, Sep 15 2005

sign

Seiichi Manyama, Table of n, a(n) for n = 0..367
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, arXiv:math/0509316 [math.NT], 2005-2006.
N. Heninger, E. M. Rains and N. J. A. Sloane, On the Integrality of n-th Roots of Generating Functions, J. Combinatorial Theory, Series A, 113 (2006), 1732-1745.

E_6^(k/12): this sequence (k=1), A289325 (k=2), A289326 (k=3), A289327 (k=4), A289328 (k=5), A289293 (k=6), A289345 (k=7), A289346 (k=8), A289347 (k=9), A289348 (k=10), A289349 (k=11).

Cf. A001160, A013973, A288851, A299503, A303055.

nmax = 20; s = 6; CoefficientList[Series[(1 - 2*s/BernoulliB[s] * Sum[DivisorSigma[s - 1, k]*x^k, {k, 1, nmax}])^(1/12), {x, 0, nmax}], x] (* Vaclav Kotesovec, Jul 02 2017 *)

G.f.: Product_{n>=1} (1-q^n)^(A288851(n)/12). - Seiichi Manyama, Jul 02 2017
a(n) ~ c * exp(2*Pi*n) / n^(13/12), where c = -Gamma(1/4)^(10/3) * Gamma(1/3)^2 / (16 * 6^(1/12) * Pi^3 * Gamma(1/12)) = -0.079329971529325538458906713053582098... - Vaclav Kotesovec, Jul 02 2017, updated Mar 05 2018
Equivalently, c = -Gamma(1/3) * Gamma(1/4)^(7/3) / (2^(23/6) * 3^(11/24) * sqrt(1 + sqrt(3)) * Pi^(5/2)). - Vaclav Kotesovec, Aug 03 2025
a(0) = 1, a(n) = -(1/n)*Sum_{k=1..n} A299503(k)*a(n-k) for n > 0. - Seiichi Manyama, Feb 27 2018
G.f.: Sum_{k>=0} A303055(k) * f(q)^k where f(q) is Sum_{k>=1} sigma_5(k)*q^k. - Seiichi Manyama, Jun 15 2018

A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

Original entry on oeis.org

504, 143388, 51180024, 20556578700, 8806299845112, 3929750661380124, 1803727445909594616, 845145871847732769804, 402283166289266872824312, 193877350835487271784566812, 94381548697864188120110027256, 46328820782943001597184984563596

Offset: 1

Seiichi Manyama, Jun 18 2017

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..367
R. E. Borcherds, Automorphic forms on O_{s+2,2}(R)^{+} and generalized Kac-Moody algebras, pp. 744-752 of Proc. Intern. Congr. Math., Vol. 2, 1994.

Cf. A288968 (k=2), A110163 (k=4), this sequence (k=6), A288471 (k=8), A289024 (k=10), A288990/A288989 (k=12), A289029 (k=14).

Cf. A008683, A013973 (E_6), A110163, A288840 (E_8/E_6), A289637.

a(n) = A013975(n^2) for n>=1.
a(n) = 12 + (1/(2*n)) * Sum_{d|n} A008683(n/d) * A288840(d).
a(n) = (1/n) * Sum_{d|n} A008683(n/d) * A289637(d). - Seiichi Manyama, Jul 09 2017
a(n) ~ exp(2*Pi*n) / n. - Vaclav Kotesovec, Mar 08 2018

A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 18, 84, 292, 630, 1512, 2408, 4680, 6813, 11340, 14652, 24528, 28574, 43344, 52920, 74896, 83538, 122634, 130340, 183960, 202272, 263736, 279864, 393120, 393775, 514332, 551880, 703136, 707310, 952560, 923552, 1198368, 1230768, 1503684, 1517040, 1989396, 1874198, 2346120, 2400216, 2948400

Offset: 0

N. J. A. Sloane, Feb 04 2017

The q-expansion of the square of this expression is given in A281371.

Multiplicative because A001158 is. - Andrew Howroyd, Jul 23 2018

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A001158, A003840, A006352, A004009, A013973, A064987, A145094, A281371, A328259.

[0] cat [n*DivisorSigma(3, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

with(gfun):
with(numtheory); M:=100;
E := proc(k) local n, t1; global M;
t1 := 1-(2*k/bernoulli(k))*add(sigma[k-1](n)*q^n, n=1..M+1);
series(t1, q, M+1); end;
e2:=E(2); e4:=E(4); e6:=E(6);
t1:=series((e2*e4-e6)/720,q,M+1);
seriestolist(t1);
# alternative program
seq(add(sigma[4](d)*phi(n/d), d in divisors(n)), n = 1..100); # Peter Bala, Jan 20 2024

Table[If[n==0, 0, n * DivisorSigma[3, n]], {n, 0, 40}] (* Indranil Ghosh, Mar 11 2017 *)
terms = 41; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(Ei[2] Ei[4] - Ei[6])/720 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

for(n=0, 40, print1(if(n==0, 0, n * sigma(n, 3)), ", ")) \\ Indranil Ghosh, Mar 11 2017

a(n) = A145094(n)/240 for n > 0. - Seiichi Manyama, Feb 04 2017
G.f.: phi_{4, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Seiichi Manyama, Feb 04 2017
a(n) = n*A001158(n) for n > 0. - Seiichi Manyama, Feb 18 2017
G.f.: x*f'(x), where f(x) = Sum_{k>=1} k^3*x^k/(1 - x^k). - Ilya Gutkovskiy, Aug 31 2017
Sum_{k=1..n} a(k) ~  Pi^4 * n^5 / 450. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-4). (End)
a(n) = Sum_{k = 1..n} sigma_4( gcd(k, n) ) = Sum_{d divides n} sigma_4(d) * phi(n/d). - Peter Bala, Jan 19 2024
a(n) = Sum_{1 <= i, j, k, l <= n} sigma_1( gcd(i, j, k, l, n) ) = Sum_{d divides n} sigma_1(d) * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 22 2024

A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 258, 6564, 66052, 390630, 1693512, 5764808, 16909320, 43066413, 100782540, 214358892, 433565328, 815730734, 1487320464, 2564095320, 4328785936, 6975757458, 11111134554, 16983563060, 25801892760, 37840199712, 55304594136, 78310985304, 110992776480

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A013955 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^8*6^1 + 2^8*3^1 + 3^8*2^1 + 6^8*1^1 = 1693512.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), A282050 (phi_{6, 1}), this sequence (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282101 (E_2*E_4^2), A013974 (E_4*E_6 = E_10).
Cf. A013955, A013666, A196288.

terms = 25;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*(E2[x]*E4[x] - E6[x])/720 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[7, n], {n, 0, 24}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^8*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n*sigma(n, 7)) \\ Andrew Howroyd, Jul 25 2018

G.f.: phi_{8, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282101(n) - A013974(n))/720. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^8 + p = A196288(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A013955(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(8) * n^9 / 9. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(7*e+7)-1)/(p^7-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-8). (End)
G.f. Sum_{k>=1} k^8*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 66, 732, 4228, 15630, 48312, 117656, 270600, 533637, 1031580, 1771572, 3094896, 4826822, 7765296, 11441160, 17318416, 24137586, 35220042, 47045900, 66083640, 86124192, 116923752, 148035912, 198079200, 244218775, 318570252, 389021400, 497449568, 594823350

Offset: 0

Seiichi Manyama, Feb 05 2017

Multiplicative because A001160 is. - Andrew Howroyd, Jul 23 2018

			a(6) = 1^6*6^1 + 2^6*3^1 + 3^6*2^1 + 6^6*1^1 = 48312.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A064987 (phi_{2, 1}), A281372 (phi_{4, 1}), this sequence (phi_{6, 1}), A282060 (phi_{8, 1}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A145095 (-q*E'_6), A008410 (E_4^2 = E_8), A282096 (E_2*E_6).
Cf. A001160, A013664, A131472.

terms = 30;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
(E4[x]^2 - E2[x]*E6[x])/1008 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)
Table[n * DivisorSigma[5, n], {n, 0, 30}] (* Amiram Eldar, Sep 06 2023 *)
nmax = 40; CoefficientList[Series[x*Sum[k^6*x^(k-1)/(1 - x^k)^2, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 01 2025 *)

a(n) = if(n < 1, 0, n * sigma(n, 5)); \\ Andrew Howroyd, Jul 23 2018

a(n) = A145095(n)/504 for n > 0.
G.f.: phi_{6, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A008410(n) - A282096(n))/1008. - Seiichi Manyama, Feb 10 2017
If p is a prime, a(p) = p^6 + p = A131472(p). - Seiichi Manyama, Feb 10 2017
a(n) = n*A001160(n) for n > 0. - Seiichi Manyama, Feb 18 2017
Sum_{k=1..n} a(k) ~ zeta(6) * n^7 / 7. - Amiram Eldar, Sep 06 2023
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^e * (p^(5*e+5)-1)/(p^5-1).
Dirichlet g.f.: zeta(s-1)*zeta(s-6). (End)
G.f. Sum_{k>=1} k^6*x^(k-1)/(1 - x^k)^2. - Vaclav Kotesovec, Aug 01 2025

A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

1, -288, -129168, -1927296, 65152656, 1535768640, 15223408704, 98001292032, 474055120080, 1870878793824, 6312358836000, 18835985199744, 50831420617152, 126257508465984, 292348744636032, 637474437331200, 1319883180896592, 2610964045674432, 4963491913583664

Offset: 0

Seiichi Manyama, Feb 06 2017

sign

The series expansion of the 12th root of the generating function gives A341801. - Peter Bala, Feb 23 2021

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A013974 (E_10).

Cf. A281374 (E_2^2), A282019 (E_2*E_4), A282096 (E_2*E_6), A282101 (E_2*E_8), this sequence (E_2*E_10), A341801.

terms = 19;
E2[x_] = 1 - 24*Sum[k*x^k/(1 - x^k), {k, 1, terms}];
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E2[x]*E4[x]*E6[x] + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 23 2018 *)

A000706 Expansion of modular function 1/E_3 (cf. A013973).

Original entry on oeis.org

1, 504, 270648, 144912096, 77599626552, 41553943041744, 22251789971649504, 11915647845248387520, 6380729991419236488504, 3416827666558895485479576, 1829682703808504464920468048, 979779820147442370107345764512

Offset: 0

N. J. A. Sloane

Ramanujan Lambert series: P(q) (see A006352), Q(q) (A004009), R(q) (A013973).

			G.f. = 1 + 504*q + 270648*q^2 + 144912096*q^3 + 77599626552*q^4 + 41553943041744*q^5 + ...

S. Ramanujan, Collected Papers of Srinivasa Ramanujan, pp. 115-7, Ed. G. H. Hardy et al., AMS Chelsea 2000, p. 317.
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).

Vaclav Kotesovec, Table of n, a(n) for n = 0..360
S. Ramanujan, On the coefficients in the expansions of certain modular functions, Proc. Royal Soc., A, 95 (1918), 144-155 [G. H. Hardy, Coll. Papers, Vol. 1, 294-305.] - Added by _N. J. A. Sloane_, Feb 21 2010

Cf. A013973, A259150.

a[ n_] := SeriesCoefficient[ 1 / (1 + Sum[ -504 DivisorSigma[ 5, k] q^k, {k, n}]), {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t2 = EllipticTheta[ 2, 0, q]^4, t3 = EllipticTheta[ 3, 0, q]^4}, 1 / (t2^3 - 33 (t2 + t3) t2 t3 + t3^3)], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {t3 = EllipticTheta[ 3, 0, q]^4, t4 = EllipticTheta[ 4, 0, q]^4}, 2 / (t3^3 - 3 (t3 - t4)^2 (t3 + t4) + t4^3) ], {q, 0, 2 n}]; (* Michael Somos, Apr 26 2015 *)
a[ n_] := SeriesCoefficient[ With[ {e1 = QPochhammer[ q]^8, e4 = 32 q QPochhammer[ q^4]^8}, QPochhammer[ q^2]^12 / ((e1 + e4) (e1^2 - 16 e1 e4 - 8 e4^2)) ], {q, 0, n}]; (* Michael Somos, Apr 26 2015 *)

{a(n) = if( n<0, 0, polcoeff( 1 / sum(k=1, n, -504*sigma(k, 5)*x^k, 1 + x * O(x^n)), n))}; /* Michael Somos, Aug 09 2007 */

{a(n) = my(A, e1, e4); if( n<0, 0, A = x * O(x^n); e1 = eta(x + A)^8; e4 = 32 * x * eta(x^4 + A)^8; polcoeff( eta(x^2 + A)^12 / ((e1 + e4) * (e1^2 - 16*e1*e4 - 8*e4^2)), n))}; /* Michael Somos, Apr 26 2015 */

Expansion of 1 / R(q) in powers of q where R() is a Ramanujan Lambert series.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = v^2*w^2 + 121*u^2*w^2 + 4096*u^2*v^2 - 8*v^3*w - 512*u*v^3 - 66*u*v*w^2 + 592*u*v^2*w - 4224*u^2*v*w. - Michael Somos, Aug 09 2007
Convolution inverse of A013973.
Asymptotics [Ramanujan]: a(n) ~ c * exp(2*Pi*n), where c = 2 / (96^2 * exp(-8*Pi/3) * Product_{j>=1} (1-exp(-4*Pi*j))^16) = 8192 * Pi^12 / (9 * Gamma(1/4)^16) = 0.943732053240742502013763912292610373458373085328537967959184338319972... . - Vaclav Kotesovec, Nov 08 2015

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 12, 36, 112, 150, 432, 392, 960, 1053, 1800, 1452, 4032, 2366, 4704, 5400, 7936, 5202, 12636, 7220, 16800, 14112, 17424, 12696, 34560, 19375, 28392, 29160, 43904, 25230, 64800, 30752, 64512, 52272, 62424, 58800, 117936, 52022, 86640, 85176, 144000, 70602

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A000203 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^3*6^2 + 2^3*3^2 + 3^3*2^2 + 6^3*1^2 = 432.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. this sequence (phi_{3, 2}), A282099 (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282018 (E_2^3), A282019 (E_2*E_4).
Cf. A000203 (sigma(n)), A064987 (n*sigma(n)), this sequence (n^2*sigma(n)), A282211 (n^3*sigma(n)).
Cf. A222171.

[0] cat [n^2*DivisorSigma(1, n): n in [1..50]]; // Vincenzo Librandi, Mar 01 2018

a[0]=0;a[n_]:=(n^2)*DivisorSigma[1,n];Table[a[n],{n,0,41}] (* Indranil Ghosh, Feb 21 2017 *)
terms = 42; Ei[n_] = 1-(2n/BernoulliB[n]) Sum[k^(n-1) x^k/(1-x^k), {k, terms}]; CoefficientList[(3*Ei[2]*Ei[4] - 2*Ei[6] - Ei[2]^3)/1728 + O[x]^terms, x] (* Jean-François Alcover, Mar 01 2018 *)

a(n) = if (n==0, 0, n^2*sigma(n)); \\ Michel Marcus, Feb 21 2017

a(n) = (3*A282019(n) - 2*A013973(n) - A282018(n))/1728.
G.f.: phi_{3, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = n^2*A000203(n) for n > 0. - Seiichi Manyama, Feb 19 2017
G.f.: Sum_{k>=1} k^3*x^k*(1 + x^k)/(1 - x^k)^3. - Ilya Gutkovskiy, May 02 2018
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(e+1)-1)/(p-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-3).
Sum_{k=1..n} a(k) ~ (Pi^2/24) * n^4. (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= i, j, k <= n} sigma_2( gcd(i, j, k, n) ).
a(n) = Sum_{1 <= i, j <= n} sigma_3( gcd(i, j, n) ).
a(n) = Sum_{d divides n} sigma_2(d) * J_3(n/d) = Sum_{d divides n} sigma_3(d) * J_2(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n). (End)

A282287 Coefficients in q-expansion of E_4*E_6^2, where E_4 and E_6 are respectively the Eisenstein series A004009 and A013973.

Original entry on oeis.org

1, -768, -19008, 67329024, 4834170816, 137655866880, 2122110676224, 21418943158272, 158760815970240, 928988742914304, 4512155542392960, 18847838706545664, 69519052583699712, 230952254655327744, 701948326302761472, 1975789128222443520

Offset: 0

Seiichi Manyama, Feb 11 2017

sign

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A004009 (E_4), A013973 (E_6), A008411 (E_4^3), A058550 (E_4^2*E_6 = E_14), this sequence (E_4*E_6^2), A282253 (E_6^3).

Cf. A282102 (E_2*E_10), A058550 (E_4*E_10), this sequence (E_6*E_10).

terms = 16;
E4[x_] = 1 + 240*Sum[k^3*x^k/(1 - x^k), {k, 1, terms}];
E6[x_] = 1 - 504*Sum[k^5*x^k/(1 - x^k), {k, 1, terms}];
E4[x]*E6[x]^2 + O[x]^terms // CoefficientList[#, x]& (* Jean-François Alcover, Feb 26 2018 *)

A282099 Coefficients in q-expansion of (E_2^2E_4 - 2E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

Original entry on oeis.org

0, 1, 36, 252, 1168, 3150, 9072, 16856, 37440, 61317, 113400, 161172, 294336, 371462, 606816, 793800, 1198336, 1420146, 2207412, 2476460, 3679200, 4247712, 5802192, 6436872, 9434880, 9844375, 13372632, 14900760, 19687808, 20511990, 28576800, 28630112, 38347776

Offset: 0

Seiichi Manyama, Feb 06 2017

Multiplicative because A001158 is. - Andrew Howroyd, Jul 25 2018

			a(6) = 1^5*6^2 + 2^5*3^2 + 3^5*2^2 + 6^5*1^2 = 9072.

Seiichi Manyama, Table of n, a(n) for n = 0..1000

Cf. A282097 (phi_{3, 2}), this sequence (phi_{5, 2}).
Cf. A006352 (E_2), A004009 (E_4), A013973 (E_6), A282208 (E_2^2*E_4), A282096 (E_2*E_6), A008410 (E_8 = E_4^2).
Cf. A001158 (sigma_3(n)), A281372 (n*sigma_3(n)), this sequence (n^2*sigma_3(n)), A282213 (n^3*sigma_3(n)).

a[0]=0;a[n_]:=(n^2)*DivisorSigma[3,n];Table[a[n],{n,0,32}] (* Indranil Ghosh, Feb 21 2017 *)
nmax = 40; CoefficientList[Series[Sum[k^5*x^k*(1 + x^k)/(1 - x^k)^3, {k, 1, nmax}], {x, 0, nmax}], x] (* Vaclav Kotesovec, Aug 02 2025 *)

a(n) = if (n==0, 0, n^2*sigma(n, 3)); \\ Michel Marcus, Feb 21 2017

G.f.: phi_{5, 2}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}.
a(n) = (A282208(n) - 2*A282096(n) + A008410(n))/1728. - Seiichi Manyama, Feb 19 2017
a(n) = n^2*A001158(n) for n > 0. - Seiichi Manyama, Feb 19 2017
Sum_{k=1..n} a(k) ~ Pi^4 * n^6 / 540. - Vaclav Kotesovec, May 09 2022
From Amiram Eldar, Oct 30 2023: (Start)
Multiplicative with a(p^e) = p^(2*e) * (p^(3*e+3)-1)/(p^3-1).
Dirichlet g.f.: zeta(s-2)*zeta(s-5). (End)
G.f.: Sum_{k>=1} k^5*x^k*(1 + x^k)/(1 - x^k)^3. - Vaclav Kotesovec, Aug 02 2025

cp's OEIS Frontend

A109817 G.f.: 12th root of Eisenstein series E_6 (cf. A013973).

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A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504*q - 16632*q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

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A281372 Coefficients in q-expansion of (E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282060 Coefficients in q-expansion of E_4*(E_2*E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282050 Coefficients in q-expansion of (E_4^2 - E_2*E_6)/1008, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A282102 Coefficients in q-expansion of E_2*E_4*E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A000706 Expansion of modular function 1/E_3 (cf. A013973).

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A282097 Coefficients in q-expansion of (3*E_2*E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

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A288851 Exponents a(1), a(2), ... such that E_6, 1 - 504q - 16632q^2 - ... (A013973) is equal to (1-q)^a(1) (1-q^2)^a(2) (1-q^3)^a(3) ... .

A282060 Coefficients in q-expansion of E_4(E_2E_4 - E_6)/720, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282102 Coefficients in q-expansion of E_2E_4E_6, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282097 Coefficients in q-expansion of (3E_2E_4 - 2*E_6 - E_2^3)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.

A282099 Coefficients in q-expansion of (E_2^2E_4 - 2E_2*E_6 + E_4^2)/1728, where E_2, E_4, E_6 are the Eisenstein series shown in A006352, A004009, A013973, respectively.