A248395 -id:A248395 - cp's OEIS frontend

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

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

A248403 Noncongruent squarefree numbers n with A248395(n)/d(n) = 1, where d(n) = A000005(n).

Original entry on oeis.org

2, 10, 58, 74, 114, 122, 130, 170, 258, 290, 314, 346, 354, 362, 370, 402, 474, 506, 586, 610, 618, 642, 714, 730, 746, 786, 826, 906, 922, 946, 962, 970, 986, 1066, 1074, 1090, 1162, 1194, 1218, 1258, 1306, 1338, 1378, 1474, 1506, 1514, 1554, 1562, 1626, 1658

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

A248404 Noncongruent squarefree numbers n with A248395(n)/d(n) = -1, where d(n) = A000005(n).

Original entry on oeis.org

26, 42, 66, 106, 186, 202, 266, 418, 498, 530, 554, 570, 634, 682, 690, 754, 762, 770, 834, 858, 874, 930, 1010, 1034, 1082, 1114, 1130, 1266, 1290, 1298, 1354, 1370, 1410, 1490, 1570, 1586, 1634, 1698, 1834, 1930, 1946, 1986, 1994, 2002, 2074, 2082, 2090, 2146

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

A248405 Noncongruent squarefree numbers n with A248395(n)/d(n) = 2, where d(n) = A000005(n).

Original entry on oeis.org

82, 282, 562, 626, 818, 914, 1042, 1106, 1138, 1202, 1426, 1442, 1578, 1618, 1778, 1802, 1874, 2114, 2258, 2338, 2410, 2482, 2506, 2562, 2642, 2674, 2714, 2866, 2874, 2938, 2954, 3322, 3498, 3506, 3602, 3810, 4314, 4354, 4458, 4562, 4826, 4930, 5026, 5258, 5322

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

A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

Original entry on oeis.org

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

Offset: 0

N. J. A. Sloane, Oct 18 2014

sign

g = q*Product_{m=1..oo} (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 every other term is zero, this one is important enough to warrant an exception.

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

The nonzero bisection is A034950, which has further information and references.
Used in A248397-A248406.
Cf. A000122 (theta_3(q)), A072068, A072069, A080917, A080918, A248395.

# 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(2),q,100);
seriestolist(%);
# Alternative with https://oeis.org/transforms.txt and the Somos Euler transform in A034950:
p8 := [2,-3,2,-2,2,-3,2,-3] ;
L := [seq(op(p8),i=1..10)] ;
EULER(%) ;
[1,op(%)] ;
[0,op(AERATE(%,1))] ; # R. J. Mathar, Nov 11 2014

QP = QPochhammer; s = q*QP[q^8]*QP[q^16]*EllipticTheta[3, 0, q^2] + O[q]^80; CoefficientList[s, q] (* Jean-François Alcover, Nov 27 2015 *)

From Seiichi Manyama, Sep 30 2018: (Start)
Let q = exp(Pi i t).
theta_3(q) = 1 + 2*q + 2*q^4 + 2*q^9 + 2*q^16 + ... .
G.f.: (theta_3(q) - theta_3(q^4))*(theta_3(q^32) - theta_3(q^8)/2)*theta_3(q^2).
a(2*n-1) = A080918(2*n-1) - A080917(2*n-1)/2 = A072069(n) - A072068(n)/2 for n > 0. (End)

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

Original entry on oeis.org

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

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.

A080966 Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.

Original entry on oeis.org

1, 2, -1, -2, 0, -4, -1, 2, -4, 2, 4, 2, 1, -2, 4, 2, 4, 0, -4, 0, -3, 4, -4, -4, 0, -2, 0, -6, 0, 2, -1, -4, 4, -4, -4, 8, 4, 6, 0, 2, -8, 0, 7, 2, 4, 2, 4, 0, 0, -6, 4, 0, -4, 0, 0, 0, 1, -6, -4, 4, -8, -2, -4, 4, 0, 2, -4, -6, 0, -2, 4, -8, 1, 2, 0, 0, 4, 4, 4, -2, 4, 6, 0, -2, 0, -4, -8, 10, 8, 8, -1, 4, 4, 2, -4, -4, -8, 6, 4, -6, 8, -6, 4, 4

Offset: 0

Michael Somos, Feb 28 2003

sign

The nonzero quadrisection of A248395.

Ramanujan theta functions: f(q) := Prod_{k>=1} (1-(-q)^k) (see A121373), phi(q) := theta_3(q) := Sum_{k=-oo..oo} q^(k^2) (A000122), psi(q) := Sum_{k=0..oo} q^(k*(k+1)/2) (A010054), chi(q) := Prod_{k>=0} (1+q^(2k+1)) (A000700).

			q + 2*q^5 - q^9 - 2*q^13 - 4*q^21 - q^25 + 2*q^29 - 4*q^33 + ...

G. C. Greubel, Table of n, a(n) for n = 0..5000
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

QP = QPochhammer; s = QP[q^2]^6/(QP[q]^2*QP[q^4]) + O[q]^100; CoefficientList[s, q] (* Jean-François Alcover, Nov 14 2015, adapted from PARI *)
QP := QPochhammer; a:=CoefficientList[Series[QP[q^2]^6/(QP[q]^2*QP[q^4]), {q, 0, 60}], q]; Table[a[[n]], {n, 1, 50}] (* G. C. Greubel, Jul 01 2018 *)

{a(n)=local(A); if(n<0, 0, A=x*O(x^n); polcoeff( eta(x^2+A)^6/eta(x+A)^2/eta(x^4+A), n))}

G.f.: Product_{k>0} (1+x^k)^2*(1-x^(2k))^3/(1+x^(2k)).
Expansion of f(-q^4)*f(q)^2 in powers of q where f(-q)=f(-q,-q^2) is a Ramanujan theta function.
Expansion of q^(-1/4)*eta(q^2)^6/(eta(q)^2*eta(q^4)) in powers of q.
Euler transform of period-4 sequence [2,-4,2,-3,...].
G.f.: Product_{k>0} (1-x^(2*k))^3*(1+x^k)^2/(1+x^(2*k)).
2*a(n) = A080964(4*n+1) = 2*A072071(4*n+1) - A072070(4*n+1).

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

cp's OEIS Frontend

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

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A248403 Noncongruent squarefree numbers n with A248395(n)/d(n) = 1, where d(n) = A000005(n).

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A248404 Noncongruent squarefree numbers n with A248395(n)/d(n) = -1, where d(n) = A000005(n).

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A248405 Noncongruent squarefree numbers n with A248395(n)/d(n) = 2, where d(n) = A000005(n).

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A248394 q-Expansion of the modular form of weight 3/2, g*theta(2) in Tunnell's notation (see Comments).

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

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A248407 Squarefree noncongruent numbers.

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A080966 Expansion of theta_4(q^2) * theta_2(q)^2/(4*q^(1/2)) in powers of q.

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Mathematica

PARI

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

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

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