cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-9 of 9 results.

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

Views

Author

N. J. A. Sloane, Oct 18 2014

Keywords

Comments

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.

Links

Crossrefs

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.

Programs

  • Maple
    # 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
  • Mathematica
    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 *)

Formula

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)

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

Views

Author

N. J. A. Sloane, Oct 18 2014

Keywords

Comments

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.

Links

Crossrefs

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

Programs

  • Maple
    # 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(%);
  • Mathematica
    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 *)
  • PARI
    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

Formula

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)

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

Intersection of A005117 and A165564.
Cf. A003273, A248394, A248395, A248397-A248406.

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

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

Views

Author

N. J. A. Sloane, Oct 20 2014

Keywords

Links

Crossrefs

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

Extensions

More terms from Amiram Eldar, Oct 13 2019
Showing 1-9 of 9 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

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