A248394 -id:A248394 - cp's OEIS frontend

A248397 Noncongruent squarefree numbers n with A248394(n)/d(n) = 1, where d(n) = A000005(n).

1, 3, 33, 51, 57, 59, 83, 139, 177, 187, 209, 211, 267, 321, 339, 345, 379, 385, 411, 451, 489, 499, 515, 555, 587, 595, 649, 659, 665, 681, 707, 803, 811, 827, 835, 899, 921, 969, 1001, 1059, 1099, 1137, 1171, 1211, 1219, 1235, 1259, 1267, 1281, 1315, 1329, 1363

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A248398 Noncongruent squarefree numbers n with A248394(n)/d(n) = -1, where d(n) = A000005(n).

Original entry on oeis.org

11, 19, 35, 67, 91, 105, 115, 123, 129, 179, 195, 201, 227, 235, 249, 273, 347, 393, 403, 419, 427, 435, 473, 483, 563, 611, 635, 683, 691, 705, 715, 739, 753, 779, 787, 795, 817, 843, 851, 993, 1051, 1115, 1121, 1123, 1177, 1209, 1265, 1347, 1401, 1435, 1441

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A248399 Noncongruent squarefree numbers n with A248394(n)/d(n) = 2, where d(n) = A000005(n).

Original entry on oeis.org

73, 155, 185, 203, 241, 281, 329, 355, 545, 553, 579, 601, 627, 641, 697, 755, 763, 785, 865, 937, 1097, 1139, 1193, 1227, 1243, 1289, 1299, 1353, 1371, 1457, 1465, 1537, 1721, 1753, 1763, 1841, 1865, 1913, 1937, 1961, 2017, 2041, 2105, 2177, 2281, 2307, 2353

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

A248400 Noncongruent squarefree numbers n with A248394(n)/d(n) = -2, where d(n) = A000005(n).

Original entry on oeis.org

17, 89, 97, 193, 217, 233, 259, 305, 377, 401, 449, 481, 497, 617, 667, 713, 745, 769, 897, 929, 955, 977, 979, 1009, 1011, 1027, 1033, 1049, 1065, 1337, 1345, 1355, 1385, 1409, 1417, 1489, 1507, 1555, 1739, 1769, 1771, 1801, 1803, 1817, 1921, 1945, 2001, 2019

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

A248401 Noncongruent squarefree numbers n with A248394(n)/d(n) = 3, where d(n) = A000005(n).

Original entry on oeis.org

43, 131, 163, 417, 491, 537, 571, 619, 849, 913, 923, 1019, 1187, 1203, 1579, 1641, 1707, 1747, 1779, 1835, 1843, 1907, 2003, 2051, 2123, 2203, 2227, 2235, 2315, 2537, 2563, 2649, 2747, 2787, 2913, 2931, 3083, 3107, 3307, 3345, 3363, 3561, 3587, 3611, 3651, 3659

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A165564, A248394, A248395, A248397-A248407.

More terms from Amiram Eldar, Oct 13 2019

A248402 Noncongruent squarefree numbers n with A248394(n)/d(n) = -3, where d(n) = A000005(n).

Original entry on oeis.org

107, 251, 283, 331, 547, 633, 643, 699, 737, 771, 883, 1041, 1147, 1163, 1307, 1459, 1483, 1497, 1523, 1531, 1619, 1627, 1699, 1793, 1883, 1915, 1923, 2049, 2179, 2251, 2283, 2361, 2363, 2411, 2427, 2433, 2443, 2467, 2539, 2651, 2843, 2971, 3091, 3147, 3187, 3203

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).

Original entry on oeis.org

0, 1, 0, 0, 0, 2, 0, 0, 0, -1, 0, 0, 0, -2, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, -1, 0, 0, 0, 2, 0, 0, 0, -4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 1, 0, 0, 0, -2, 0, 0, 0, 4, 0, 0, 0, 2, 0, 0, 0, 4, 0, 0, 0, 0, 0, 0, 0, -4, 0, 0, 0, 0

Offset: 0

N. J. A. Sloane, Oct 18 2014

sign

g = Product_{m>=1} (1-q^(8*m))*(1-q^(16*m)),
theta(t) = Sum_{n=-oo..oo} (q^(t*n^2)).
Although the OEIS does not normally include sequences in which only every fourth term is nonzero, this one is important enough to warrant an exception.

G. C. Greubel, Table of n, a(n) for n = 0..5000
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

The nonzero quadrisection is A080966, which has further information and references.
Cf. A248394.
Used in A248397-A248406.

# This produces a list of the first 100 terms:
g:=q*mul((1-q^(8*m))*(1-q^(16*m)),m=1..30);
g:=series(g,q,100);
th:=t->series( add(q^(t*n^2),n=-50..50), q, 100);
series(g*th(4),q,100);
seriestolist(%);

QP := QPochhammer; a:= CoefficientList[Series[QP[q^8]*QP[q^16]* EllipticTheta[3, 0, q^4], {q, 0, 60}], q]; Join[{0}, Table[a[[n]], {n, 1, 50}]] (* G. C. Greubel, Jul 02 2018 *)

my(q='q+O('q^80)); A = eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)); concat([0], Vec(A)) \\ G. C. Greubel, Jul 02 2018

From G. C. Greubel, Jul 02 2018: (Start)
Expansion of eta(q^8)*eta(q^16)*theta_{3}(0, q^4)/q in powers of q.
Expansion of eta(q^8)^6/(q*eta(q^4)^2*eta(q^16)). (End)

A248406 Noncongruent squarefree numbers n with A248395(n)/d(n) = -2, where d(n) = A000005(n).

Original entry on oeis.org

146, 178, 274, 322, 466, 938, 994, 1002, 1234, 1394, 1498, 1714, 1866, 1906, 2066, 2098, 2162, 2194, 2386, 2578, 2586, 2786, 2794, 2962, 3002, 3034, 3218, 3266, 3346, 3658, 3682, 3914, 3986, 4090, 4130, 4594, 4738, 4786, 4946, 5170, 5210, 5234, 5266, 5402, 5458

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Cf. A000005, A003273, A248394, A248395, A248397-A248406.

More terms from Amiram Eldar, Oct 13 2019

A034950 Expansion of eta(8z)eta(16z)theta_3(2z).

Original entry on oeis.org

1, 2, 0, 0, 1, -2, 0, 0, -4, -2, 0, 0, -3, 0, 0, 0, 4, -4, 0, 0, 0, 6, 0, 0, 1, 4, 0, 0, 4, 2, 0, 0, 0, -2, 0, 0, 4, -2, 0, 0, -3, 2, 0, 0, -4, -4, 0, 0, -4, 2, 0, 0, -8, -6, 0, 0, 8, -4, 0, 0, 1, -4, 0, 0, -4, 6, 0, 0, 0, 2, 0, 0, 0, -2, 0, 0, 4, 8, 0, 0, 0, 6, 0, 0, 5, -2, 0, 0, 4, -2, 0, 0, 8, 4, 0, 0, -4, -8, 0, 0, -4, 8, 0, 0, 4

Offset: 0

N. J. A. Sloane

sign

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

			G.f. = 1 + 2*x + x^4 - 2*x^5 - 4*x^8 - 2*x^9 - 3*x^12 + 4*x^16 - 4*x^17 + ...
G.f. = q + 2*q^3 + q^9 - 2*q^11 - 4*q^17 - 2*q^19 - 3*q^25 + 4*q^33 - ...

Seiichi Manyama, Table of n, a(n) for n = 0..10000 (terms 0..1000 from G. C. Greubel)
Ken Ono and Christopher Skinner, Fourier Coefficients of Half-Integral Weight Modular Forms Modulo l, Ann. Math., 147 (1998), 453-470.
Michael Somos, Introduction to Ramanujan theta functions
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.
Eric Weisstein's World of Mathematics, Ramanujan Theta Functions

Cf. A072068, A072069, A080963.

A bisection of A248394.

a[ n_] := SeriesCoefficient[ EllipticTheta[ 3, 0, x] EllipticTheta[ 2, 0, x] EllipticTheta[ 2, Pi/4, x] / Sqrt[8 x], {x, 0, n}]; (* Michael Somos, Feb 18 2015 *)
QP = QPochhammer; s = QP[q^2]^5*(QP[q^8]/(QP[q]^2*QP[q^4])) + O[q]^105; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

{a(n) = local(A); if( n<0, 0, A = x * O(x^n); polcoeff( eta(x^2 + A)^5 * eta(x^8 + A) / (eta(x + A)^2 * eta(x^4 + A)), n))}; /* Michael Somos, Feb 16 2006 */

Euler transform of period 8 sequence [2, -3, 2, -2, 2, -3, 2, -3, ...]. - Michael Somos, Feb 16 2006
Expansion of q^(-1/2) * eta(q^2)^5 * eta(q^8) / (eta(q)^2 * eta(q^4)) in powers of q. - Michael Somos, Feb 16 2006
Expansion of psi(x)^2 * psi(-x^2) = phi(x) * psi(x^2) * psi(-x^2) = phi(x) * psi(x^4) * phi(-x^4) in powers of x where phi(), psi() are Ramanujan theta functions. - Michael Somos, Feb 18 2015
G.f.: Product_{k>0} (1 + x^k)^2 * (1 - x^(2*k))^3 * (1 + x^(4*k)). - Michael Somos, Feb 16 2006
2 * a(n) = A080963(2*n + 1). a(4*n + 2) = a(4*n + 3) = 0. - Michael Somos, Feb 18 2015
a(n) = A072069(n+1) - A072068(n+1)/2. - Seichi Manymama, Sep 30 2018

A248407 Squarefree noncongruent numbers.

Original entry on oeis.org

1, 2, 3, 10, 11, 17, 19, 26, 33, 35, 42, 43, 51, 57, 58, 59, 66, 67, 73, 74, 82, 83, 89, 91, 97, 105, 106, 107, 113, 114, 115, 122, 123, 129, 130, 131, 139, 146, 155, 163, 170, 177, 178, 179, 185, 186, 187, 193, 195, 201, 202, 203, 209

Offset: 1

N. J. A. Sloane, Oct 20 2014

nonn

Amiram Eldar, Table of n, a(n) for n = 1..2580 (terms below 10^4)
J. B. Tunnell, A classical Diophantine problem and modular forms of weight 3/2, Invent. Math., 72 (1983), 323-334.

Intersection of A005117 and A165564.

Cf. A003273, A248394, A248395, A248397-A248406.

cp's OEIS Frontend

A248397 Noncongruent squarefree numbers n with A248394(n)/d(n) = 1, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248398 Noncongruent squarefree numbers n with A248394(n)/d(n) = -1, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248399 Noncongruent squarefree numbers n with A248394(n)/d(n) = 2, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248400 Noncongruent squarefree numbers n with A248394(n)/d(n) = -2, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248401 Noncongruent squarefree numbers n with A248394(n)/d(n) = 3, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248402 Noncongruent squarefree numbers n with A248394(n)/d(n) = -3, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A248395 q-Expansion of the modular form of weight 3/2, g*theta(4) in Tunnell's notation (see Comments).

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

Maple

Mathematica

PARI

Formula

A248406 Noncongruent squarefree numbers n with A248395(n)/d(n) = -2, where d(n) = A000005(n).

Views

Author

Keywords

Links

Crossrefs

Extensions

A034950 Expansion of eta(8z)*eta(16z)*theta_3(2z).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

A248407 Squarefree noncongruent numbers.

Views

Author

Keywords

Links

Crossrefs

A034950 Expansion of eta(8z)eta(16z)theta_3(2z).