A007332 -id:A007332 - cp's OEIS frontend

A007659 Primes p such that Ramanujan number tau(p) is divisible by p.

Original entry on oeis.org

2, 3, 5, 7, 2411, 7758337633

Offset: 1

N. J. A. Sloane, Robert G. Wilson v

Primes at which cusp form Delta_12 (see A007332) is not ordinary.

a(5) was found by Newman (1972). - Amiram Eldar, Jan 06 2025

Morris Newman, A table of tau(p) modulo p, p prime, 3 <= p <= 16067, National Bureau of Standards, 1972.
Joe Roberts, Lure of the Integers, Math. Assoc. America, 1992, p. 275.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Fernando Q. Gouvêa, Non-ordinary primes: a story, Experimental Mathematics 6(3) (1997), 195-205; alternative link.
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 (mod p). Journal of Integer Sequences, Vol. 13 (2010), Article 10.7.4.
Nik Lygeros and Olivier Rozier, A new solution for the equation tau(p)=0 mod p. Number Theory mailing list (NMBRTHRY), 2010.
Morris Newman, A table of tau(p) modulo p, p prime, 3 <= p <= 16067, Review, Mathematics of Computation, Vol. 27, No. 121 (1973), pp. 215-216.

Cf. A000594, A007332. A proper subset of A063938.

Select[ Prime[ Range[ 5133]], Mod[ RamanujanTau[ # ], # ] == 0 &] (* Dean Hickerson, Jan 03 2003 *)
Select[Prime[Range[400]],Divisible[RamanujanTau[#],#]&] (* The program generates the first 5 terms of the sequence. *) (* Harvey P. Dale, Jun 06 2022 *)

a(6) = 7758337633 from N. Lygeros and O. Rozier, Mar 16 2010. - N. J. A. Sloane, Mar 16 2010

Edited by Max Alekseyev, Jul 11 2010

A037150 Fourier coefficients of Eisenstein series of degree 2 and weight 6 when evaluated at Gram(A_2)*z.

Original entry on oeis.org

1, 0, -1512, 44352, 449064, 6519744, 47263608, 257027904, 1115041032, 4093040448, 13000566096, 37057027392, 96945887304, 232758852480, 526296318912, 1128198915648, 2286101175624, 4451375005056, 8386154766360, 15131349955008, 26614555499952

Offset: 0

N. J. A. Sloane

sign

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag, Third Ed., 1998.
Helmut Klingen, Introductory Lectures on Siegel Modular Forms, Cambrifge, 1990, p. 123.

N. J. A. Sloane, Table of n, a(n) for n = 0..100
N. J. A. Sloane, Notes on Two-dimensional Theta Series of Lattices (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".
Index entries for sequences related to Eisenstein series

Cf. A004016, A007332.

# Maple code from N. J. A. Sloane, Dec 12 2020. Will also be useful for related sequences.
# get standard theta series in Maple. First set max degree, maxd.
maxd:=201:
# get th2, th3, th4 = Jacobi theta constants out to degree maxd (Ref. Conway-Sloane, p. 102)
temp0:=trunc(evalf(sqrt(maxd)))+2:
a:=0: for i from -temp0 to temp0 do a:=a+q^( (i+1/2)^2): od:
th2:=series(a,q,maxd);
a:=0: for i from -temp0 to temp0 do a:=a+q^(i^2): od:
th3:=series(a,q,maxd);
th4:=series(subs(q=-q,th3),q,maxd);
# get Dedekind eta function
a:=q^(1/24) : for m from 1 to maxd do a:=a*(1-q^m); od:
eta:=a;
# get phi0 and phi1 (Ref. Conway-Sloane, p. 103)
phi0:=series( subs(q=q^2,th2)*subs(q=q^6,th2)+subs(q=q^2,th3)*subs(q=q^6,th3), q, maxd );
phi1:=series( subs(q=q^2,th2)*subs(q=q^6,th3)+subs(q=q^2,th3)*subs(q=q^6,th2), q, maxd );
# get delta12 (Ref. Conway-Sloane, p. 204, where it is called Delta_6)
delta12:=series((subs(q=q^3,eta)*eta)^6,q,maxd);
delta12:=series(subs(q=q^2,delta12),q,maxd);
# To get the present sequence: (Ref. Sloane notebook pages)
x:=phi0; y:=delta12;
w1:= x^12-72*x^6*y-1728*y^2; w1s:=series(w1,q,maxd); w2:=subs(q=sqrt(t),w1s); w3:=series(w2,t,101);
w4:=seriestolist(w3); # A037150

x^12-72*x^6*y-1728*y^2, x = phi_0(z), y = Delta_12(z). Here phi_0(z) is the theta series of the A_2 lattice - see A004016, and Delta_12 is the 12-dimensional cusp form given in A007332.

Entry revised by N. J. A. Sloane, Dec 12 2020

A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

Original entry on oeis.org

1, -3, 0, 2, 9, 0, -22, 0, 0, 26, -6, 0, 25, -27, 0, -46, 0, 0, 26, 66, 0, -22, 0, 0, -45, 0, 0, 0, -78, 0, 74, 18, 0, 122, 0, 0, -46, -75, 0, -142, 81, 0, 0, 0, 0, -44, 138, 0, 2, 0, 0, 194, 0, 0, -214, -78, 0, 0, -198, 0, 121, 0, 0, 146, 66, 0, 52, 0, 0, -22

Offset: 0

N. J. A. Sloane

sign

Number 28 of the 74 eta-quotients listed in Table I of Martin (1996).

Expansion of a newform level 12 weight 3 and character [0, 1].

			G.f. = 1 - 3*x + 2*x^3 + 9*x^4 - 22*x^6 + 26*x^9 - 6*x^10 + 25*x^12 - 27*x^13 + ...
G.f. = q - 3*q^3 + 2*q^7 + 9*q^9 - 22*q^13 + 26*q^19 - 6*q^21 + 25*q^25 + ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000
Author?, Eta Products and Quotients which are Newforms. [Broken link?]
M. Koike, On McKay's conjecture, Nagoya Math. J., 95 (1984), 85-89.
Y. 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

Cf. A007332, A152243, A209939.

A := Basis( CuspForms( Gamma1(12), 3), 140); A[1] - 3*A[3]; /* Michael Somos, May 17 2015 */

a[ n_] := SeriesCoefficient[ (QPochhammer[ x] QPochhammer[ x^3])^3, {x, 0, n}]; (* Michael Somos, May 17 2015 *)

{a(n) = my(A); if( n<0, 0, A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^3, n))}; /* Michael Somos, Jun 14 2007 */

Euler transform of period 3 sequence [ -3, -3, -6, ...]. - Michael Somos, Feb 13 2006
a(n) = b(2*n + 1) where b() is multiplicative with b(2^e) = 0^e, b(3^e) = (-3)^e, b(p^e) = (1 + (-1)^e) / 2 * p^e if p == 5 (mod 6), b(p^e) = b(p) * b(p^(e-1)) - p^2 * b(p^(e-2)) otherwise. - Michael Somos, Feb 13 2006
G.f.: (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^3.
G.f.: Sum_{k>=0} a(k) * q^(2*k + 1) = (1/2) * Sum_{u, v in Z} (u*u - 3*v*v) * q^(u*u + 3*v*v). - Michael Somos, Jun 14 2007
G.f. is a period 1 Fourier series which satisfies f(-1 / (12 t)) = 12^(3/2) (t/i)^3 f(t) where q = exp(2 Pi i t). - Michael Somos, Nov 16 2008
a(3*n + 1) = -3 * a(n). a(3*n + 2) = 0. a(3*n) = A152243(n). - Michael Somos, Mar 09 2012
a(n) = (-1)^n * A209939(n). - Michael Somos, Mar 16 2012
Convolution square is A007332. - Michael Somos, Nov 16 2008

A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, 6, -27, -92, 390, -162, -64, -1320, 729, 2340, -948, 2484, -5098, -384, -10530, 3856, 28386, 4374, -8620, -35880, 1728, -5688, -15288, 35640, 73975, -30588, -19683, 5888, 36510, -63180, -276808, 192096, 25596, 170316, -24960, -67068, 268526, -51720, 137646

Offset: 1

Michael Somos, Jan 20 2008

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q + 6*q^2 - 27*q^3 - 92*q^4 + 390*q^5 - 162*q^6 - 64*q^7 - 1320*q^8 + ...

Masao Koike, Modular forms on non-compact arithmetic triangle groups, Unpublished manuscript [Extensively annotated with OEIS A-numbers by N. J. A. Sloane, Feb 14 2021. I wrote 2005 on the first page but the internal evidence suggests 1997.] See page 30, line -3.
LMFDB, Newform orbit 3.8.a.a.

Cf. A007332, A008653.

Basis( CuspForms( Gamma0(3), 8), 40)[1]; /* Michael Somos, Oct 12 2015 */

a[ n_] := SeriesCoefficient[ q (QPochhammer[ q] QPochhammer[ q^3])^6 ((QPochhammer[ q]^3 + 9 q QPochhammer[ q^9]^3) / QPochhammer[ q^3])^2, {q, 0, n}]; (* Michael Somos, May 28 2013 *)

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 * sum(k=1, n, 12 * (sigma(3*k) - 3 * sigma(k)) * x^k, 1 + A), n))};

CuspForms( Gamma0(3), 8, prec=40).0; # Michael Somos, May 28 2013

Expansion of (eta(q) * eta(q^3))^6 * ((eta(q)^3 + 9 * eta(q^9)^3) / eta(q^3))^2 in powers of q.
a(n) is multiplicative with a(3^e) = (-27)^e, a(p^e) = a(p) * a(p^(e-1)) - p^7 * a(p^(e-2)) unless p = 3.
G.f. A(x) satisfies 0 = f(A(x), A(x^2), A(x^4)) where f(u, v, w) = u^4*w + 512*u^3*w^2 + 131072*u^2*w^3 + 16777216*u*w^4 - 24*u^3*v*w - 9216*u^2*v*w^2 - 1572864*u*v*w^3 + 288*u^2*v^2*w + 73728*u*v^2*w^2 - u^2*v^3 - 1984*w*v^3*u - 65536*w^2*v^3 + 12*v^4*u + 3072*w*v^4 - 36*v^5.
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t).
G.f.: x * (Product_{k>0} (1 - x^k) * (1 - x^(3*k)))^6 * (Sum_{j,k in Z} x^(j*j + j*k + k*k))^2.
Convolution of A007332 and A008653.

A004033 Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice.

Original entry on oeis.org

1, 0, 720, 13440, 97200, 455040, 1714320, 4821120, 12380400, 29043840, 58980960, 114076800, 219310320, 367338240, 621878400, 1037727360, 1583679600, 2401816320, 3747180240, 5232470400, 7551983520, 10938261120, 14715224640, 19930775040, 28073386800, 35727920640

Offset: 0

N. J. A. Sloane

Also theta series of 16-dimensional lattice (SL(2,9) Y SL(2,9)).(C2 x C2). - John Cannon, Jan 10 2007
Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).
Ramanujan theta functions: f(q) (see A121373), phi(q) (A000122), psi(q) (A010054), chi(q) (A000700).

			G.f. = 1 + 720*x^2 + 13440*x^3 + 97200*x^4 + 455040*x^5 + 1714320*x^6 + 4821120*x^7 + ...
G.f. = 1 + 720*q^4 + 13440*q^6 + 97200*q^8 + 455040*q^10 + 1714320*q^12 + 4821120*q^14 + ...

J. H. Conway and N. J. A. Sloane, "Sphere Packings, Lattices and Groups", Springer-Verlag.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

G. C. Greubel, Table of n, a(n) for n = 0..1000
Walter Feit, Some lattices over Q(sqrt(-3)), J. Algebra 52 (1978), no. 1, 248-263.
Michael Somos, Introduction to Ramanujan theta functions
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A004016, A007332, A037150.

// Definition for lattice (SL(2,9) Y SL(2,9)).(C2 x C2), from John Cannon
LatticeWithBasis(16, \[ 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0,
0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1 ], MatrixRing(IntegerRing(), 16) ! \[
4, 1, 2, 1, 1, 2, 1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 4, 2, 1, 1, 1, 2,
1, -1, 1, 1, 0, 1, 0, 1, 1, 2, 2, 4, 0, 1, 2, 2, 1, 1, 1, 2, 1, 0, 1,
1, 2, 1, 1, 0, 4, 1, 1, 1, 2, 0, 2, 1, 1, 2, 1, 1, 0, 1, 1, 1, 1, 4,
1, 1, 1, 1, 0, 0, 0, 1, 2, 2, 2, 2, 1, 2, 1, 1, 4, 1, 1, 2, 1, 2, 2,
0, 2, 1, 2, 1, 2, 2, 1, 1, 1, 4, 1, 0, 2, 2, 1, 1, 0, 1, 1, 1, 1, 1,
2, 1, 1, 1, 4, 1, 2, 2, 2, 1, 1, 1, 0, 1, -1, 1, 0, 1, 2, 0, 1, 4, 1,
1, 2, 1, 2, 1, 1, 1, 1, 1, 2, 0, 1, 2, 2, 1, 4, 2, 2, 1, 0, 0, -1, 2,
1, 2, 1, 0, 2, 2, 2, 1, 2, 4, 2, 0, 0, 1, 1, 1, 0, 1, 1, 0, 2, 1, 2,
2, 2, 2, 4, 1, 0, -1, 1, 1, 1, 0, 2, 1, 0, 1, 1, 1, 1, 0, 1, 4, 1, 1,
1, 2, 0, 1, 1, 2, 2, 0, 1, 2, 0, 0, 0, 1, 4, 2, 1, 2, 1, 1, 1, 2, 1,
1, 1, 1, 0, 1, -1, 1, 2, 4, 1, 1, 1, 2, 0, 2, 2, 1, 0, 1, -1, 1, 1, 1,
1, 1, 4 ])

// Definition for lattice A_2 tensor E_8, from John Cannon
A := Lattice("A", 2);
B := Lattice("E", 8);
L := TensorProduct(A, B);
T := ThetaSeries(L, 16);

A := Basis( ModularForms( Gamma0(3), 8), 26); A[1] + 720*A[3]; /* Michael Somos, Feb 01 2017 */

a[ n_] := SeriesCoefficient[ With[ {a1 = (QPochhammer[ x]^3 + 9 x QPochhammer[ x^9]^3) / QPochhammer[ x^3]}, a1^2 (a1^6 - 48 x QPochhammer[ x]^6 QPochhammer[ x^3]^6)], {x, 0, n}]; (* Michael Somos, Feb 01 2017 *)

{a(n) = if( n<0, 0, my(A, a1); A = x * O(x^n); a1 = (eta(x + A)^3 + 9*x * eta(x^9 + A)^3) / eta(x^3 + A); polcoeff( a1^2 * (a1^6 - 48*x * eta(x + A)^6 * eta(x^3 + A)^6), n))}; /* Michael Somos, Feb 01 2017 */

Theta series is x^8-48*x^2*y, x = phi_0(z) (see A004016), y = Delta_12(z) (see A007332) in the notation of SPLAG, Chap. 4. See A037150 for Maple code.
Expansion of a(x)^2 * (a(x)^6 - 48*x * f(-x)^6 * f(-x^3)^6) in powers of x where a() is a cubic AGM theta function and f() is a Ramanujan theta function. - Michael Somos, Feb 01 2017
G.f. is a period 1 Fourier series which satisfies f(-1 / (3 t)) = 81 (t/i)^8 f(t) where q = exp(2 Pi i t). - Michael Somos, Feb 01 2017

A037190 Theta series of lattice A_2 tensor D12+ (dimension 24, min norm 4).

Original entry on oeis.org

1, 0, 792, 16704, 573480, 6289344, 47367288, 256779072, 1116635400, 4092791616, 12993723216, 37059359040, 96956352072, 232789209984, 526238331840, 1128121363008, 2286165374280, 4451600668032, 8386296787224, 15130668404160, 26614226534832, 45684867221568

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Notes on Two-dimensional Theta Series of Lattices (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".

G.f.: phi_0(z)^12 - 72*phi_0(z)^6*Delta_12(z) + 576*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 16 2020

More terms from Sean A. Irvine, Dec 16 2020

A037191 Theta series of lattice A_2 tensor E_7^2+ (dimension 28, min norm 4).

Original entry on oeis.org

1, 0, 756, 17472, 700812, 13487040, 143699220, 1061439552, 6038839548, 27868766016, 109814205816, 378998996544, 1173180111852, 3325312497024, 8715279904416, 21337957069632, 49450420013148, 108741984430464, 228329201738484, 461698406495424, 899506102392072

Offset: 0

N. J. A. Sloane

nonn

N. J. A. Sloane, Notes on Two-dimensional Theta Series of Lattices (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".

G.f.: phi_0(z)^14 - 84*phi_0(z)^8*Delta_12(z) + 1008*phi_0(z)^2*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 16 2020

More terms from Sean A. Irvine, Dec 16 2020

A037212 Theta series of lattice A_2 tensor A_15+ (dimension 30, det 3^15, min. norm 4).

Original entry on oeis.org

1, 0, 720, 17640, 726570, 17236728, 223847280, 1928224440, 12500672580, 65073916440, 284229347328, 1080456070680, 3650778821970, 11191823745120, 31612753519200, 83017908015624, 204808695842610, 479043914067360, 1065816689968080, 2271147404056200

Offset: 0

N. J. A. Sloane

N. J. A. Sloane, Notes on Two-dimensional Theta Series of Lattices (Notes on some joint work with Eric M. Rains), pages 96-115, circa Jun 08 1998, of N. J. A. Sloane's notebook "Lattices Volume 79".

Cf. A004016, A007332.

G.f.: phi_0(z)^15 - 90*phi_0(z)^9*Delta_12(z) + 1260*phi_0(z)^3*Delta_12(z)^2 where phi_0(z) is the theta series of the A_2 lattice given in A004016, and Delta_12 is the 12-dimensional cusp form given in A007332. - Sean A. Irvine, Dec 17 2020

More terms from Sean A. Irvine, Dec 17 2020

A128486 Expansion of ((b(q)c(q))^3 - 8(b(q^2)*c(q^2))^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.

Original entry on oeis.org

1, -14, 9, 52, 6, -126, -40, 136, 81, -84, -564, 468, 638, 560, 54, -2480, 882, -1134, -556, 312, -360, 7896, -840, 1224, -3089, -8932, 729, -2080, 4638, -756, 4400, 10528, -5076, -12348, -240, 4212, -2410, 7784, 5742, 816, -6870, 5040, 9644, -29328, 486, 11760, -18672, -22320

Offset: 1

Michael Somos, Mar 04 2007

Cubic AGM theta functions: a(q) (see A004016), b(q) (A005928), c(q) (A005882).

			G.f. = q - 14*q^2 + 9*q^3 + 52*q^4 + 6*q^5 - 126*q^6 - 40*q^7 + 136*q^8 + ...

Cf. A007332.

A := Basis( CuspForms( Gamma1(6), 6), 49); A[1] - 14*A[2] + 9*A[3]; /* Michael Somos, Feb 19 2015 */

{a(n) = my(A); if( n<1, 0, n--; A = x * O(x^n); polcoeff( (eta(x + A) * eta(x^3 + A))^6 - 8*x * (eta(x^2 + A) * eta(x^6+A))^6, n))};

{a(n) = my(A, A1, A2); if( n<1, 0, n--; A = x * O(x^n); A1 = eta(x + A) * eta(x^2 + A); A2 = eta(x^3 + A) * eta(x^6 + A); polcoeff( A1^5*A2 - 9*x * A1*A2^5, n))};

Expansion of b(q) * b(q^2) * c(q) * c(q^2) * (b(q) * b(q^2) - c(q) * c(q^2)) / 9 in powers of q where b(), c() are cubic AGM theta functions.
Expansion of (eta(q) * eta(q^3))^6 - 8*(eta(q^2) * eta(q^6))^6 in powers of q.
Expansion of eta(q) * eta(q^2) * eta(q^3) * eta(q^6) * ((eta(q) * eta(q^2))^4 - 9*(eta(q^3) * eta(q^6))^4) in powers of q.
G.f.: x * Product_{k>0} (1 - x^k)^6 * (1 - x^(3*k))^6 - 8 * x^2 * Product_{k>0} (1 - x^(2*k))^6 * (1 - x^(6*k))^6.
G.f. is a period 1 Fourier series which satisfies f(-1 / (6*t)) = -216 (t/i)^6 f(t) where q = exp(2 Pi i t).
a(2*n) = A007332(2*n) - 8 * A007332(n). a(2*n + 1) = A007332(2*n + 1).

cp's OEIS Frontend

A007659 Primes p such that Ramanujan number tau(p) is divisible by p.

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A037150 Fourier coefficients of Eisenstein series of degree 2 and weight 6 when evaluated at Gram(A_2)*z.

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A030208 Expansion of q^(-1/2) * (eta(q) * eta(q^3))^3 in powers of q.

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A136747 Expansion of a(q)^2 * (b(q) * c(q) / 3)^3 in powers of q where a(), b(), c() are cubic AGM theta functions.

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Magma

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A004033 Theta series of lattice A_2 tensor E_8 (dimension 16, det. 6561, min. norm 4). Also theta series of Eisenstein version of E_8 lattice.

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Magma

Magma

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A037190 Theta series of lattice A_2 tensor D12+ (dimension 24, min norm 4).

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A037191 Theta series of lattice A_2 tensor E_7^2+ (dimension 28, min norm 4).

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A128486 Expansion of ((b(q)c(q))^3 - 8(b(q^2)*c(q^2))^3) / 27 in powers of q where b(), c() are cubic AGM theta functions.