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 1-7 of 7 results.

A179982 Values of the genus g for which there exists a compact Riemann surface of genus g admitting an automorphism group of order 84(g-1), the maximum possible, also known as the Hurwitz bound.

Original entry on oeis.org

3, 7, 14, 17, 118, 129, 146, 385, 411, 474, 687, 769, 1009, 1025, 1459, 1537, 2091, 2131, 2185, 2663, 3404, 4369, 4375, 5433, 5489, 6553, 7201, 8065, 8193, 8589, 11626, 11665
Offset: 1

Views

Author

Richard Chapling (rc476(AT)cam.ac.uk), Aug 04 2010

Keywords

Comments

It is known that the sequence is infinite, and also that the sequence of g for which the Hurwitz bound is not attained is infinite. No generating formula is known.
The group in question is a Hurwitz group, and so is generated by x,y with x^2 = y^3 = (xy)^7 = 1.
For k in A343821 (not in A343822), k!/168 + 1 is in this sequence since the alternating group A_k is a Hurwitz group. In particular, k!/168 + 1 is a term for all k >= 168. - Jianing Song, Jul 13 2021

Examples

			g=3 is satisfied by the Klein quartic x^3 * y + y^3 * z + z^3 * x = 0; the group is isomorphic to PSL(2,Z_7), the projective special linear group of 2 X 2 matrices with entries modulo 7.

References

  • Marston Conder, Hurwitz groups: a brief survey. Bull. Amer. Math. Soc. (N.S.) 23 (1990), no. 2, 359-370.
  • Gareth A. Jones and David Singerman, Complex Functions: An algebraic and geometric viewpoint, Cambridge University Press, 1987, pp. 263-266.
  • E. B. Vinber and O. V. Shvartsman, Riemann surfaces, Journal of Mathematical Sciences, 14, #1 (1980), 985-1020. Riemann surfaces, Algebra, Topologiya, Geometriya, Vol. 16 (Russian), pp. 191-245, 247 (errata insert), VINITI , Moscow, 1978.

Links

Crossrefs

Cf. A343821, A343822, A346293.

Programs

  • Magma
    G:=Group; L:=LowIndexNormalSubgroups(G,86016); for j in L do print (j`Index)/84+1; end for; // Bradley Brock, Nov 25 2012

Extensions

Corrected by Bradley Brock, Oct 01 2012
Entry revised by N. J. A. Sloane, Nov 25 2012

A347368 Number of signatures of Fuchsian groups leading to automorphism groups of Riemann surfaces of genus n.

Original entry on oeis.org

20, 41, 56, 65, 75, 98, 92, 135, 168, 145, 167, 222, 183, 254, 283, 281, 277, 398, 337, 436, 441, 391, 499, 637, 542, 638, 731, 689, 736, 921, 805, 950, 1019, 1013, 1150, 1346, 1140, 1325, 1518, 1520, 1535, 1805, 1670, 1946, 2084, 1950, 2167
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

This includes signatures leading to subgroups of the full automorphism group.
Breuer's book erroneously gives a(33) = 949. (See errata.)

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347369, A347370, A347371, A347372, A347373.

Formula

a(n) = A347369(n) + A347370(n).

A347369 Number of signatures of Fuchsian groups of orbit genus 0 leading to automorphism groups of Riemann surfaces of genus n.

Original entry on oeis.org

18, 34, 47, 51, 63, 72, 74, 102, 130, 103, 128, 158, 136, 178, 200, 194, 197, 272, 235, 289, 299, 241, 337, 418, 354, 402, 477, 423, 471, 567, 503, 577, 618, 596, 704, 816, 672, 763, 903, 875, 891, 1028, 954, 1097, 1187, 1055, 1221
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

This includes signatures leading to subgroups of the full automorphism group.
Breuer's book erroneously gives a(33) = 576. (See errata.)

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347368, A347370, A347371, A347372, A347373.

A347370 Number of signatures of Fuchsian groups of positive orbit genus leading to automorphism groups of Riemann surfaces of genus n.

Original entry on oeis.org

2, 7, 9, 14, 12, 26, 18, 33, 38, 42, 39, 64, 47, 76, 83, 87, 80, 126, 102, 147, 142, 150, 162, 219, 188, 236, 254, 266, 265, 354, 302, 373, 401, 417, 446, 530, 468, 562, 615, 645, 644, 777, 716, 849, 897, 895, 946
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

This includes signatures leading to subgroups of the full automorphism group.

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347368, A347369, A347371, A347372, A347373.

A347371 Number of isomorphism types of automorphism groups of Riemann surfaces of genus n.

Original entry on oeis.org

19, 37, 44, 64, 59, 86, 65, 154, 119, 118, 98, 206, 99, 176, 139, 346, 117, 290, 136, 368, 187, 193, 171, 621, 184, 276, 306, 483, 187, 404, 189, 1014, 255, 332, 253, 880, 205, 381, 341, 1163, 244, 549, 244, 788, 436, 401, 273
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

This includes subgroups of the full automorphism group.
Breuer's book erroneously gives a(33) = 1013. (See errata.)

Examples

			The 19 automorphism groups for Riemann surfaces of genus 2 are the trivial group, C2, C3, C4, C2 X C2, C5, C6, S3, Q8, C8, D8, C10, C6 . C2, C2 X C6, D12, QD16, SL_2(3), (C2 X C6) . C2, and GL_2(3). [Breuer, Table 9 on p. 77]

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347368, A347369, A347370, A347372, A347373.

A347372 Number of signature-group pairs for Riemann surfaces of genus n.

Original entry on oeis.org

21, 49, 64, 93, 87, 148, 108, 268, 226, 232, 201, 453, 229, 408, 386, 733, 337, 791, 425, 941, 628, 718, 625, 1695, 715, 1101, 1147, 1642, 930, 1786, 1048, 2844, 1444, 1848, 1495, 3452, 1500, 2424, 2192, 4192, 2000, 3585, 2220, 4193, 3211, 3638, 2814
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

This includes subgroups of the full automorphism group.
Breuer's book erroneously gives a(33) = 2843. (See errata.)

Examples

			There are 20 signatures for genus 2. Of these, the signature (0; 2, 2, 3, 3) leads to both C6 and S3. Thus the total number of signature-group pairs is 21.

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347368, A347369, A347370, A347371, A347373.

A347373 Number of Aut(G)-orbits on G-characters that come from Riemann surfaces of genus n.

Original entry on oeis.org

21, 55, 73, 116, 105, 208, 141, 428, 335, 424, 329, 952, 365, 924, 789, 1834, 742, 2119, 936, 3365, 1762, 2694, 1812, 7274, 2058, 5109, 4024, 9812, 3706, 10258, 4404, 18905, 7664, 13482, 8041, 31541, 8473, 21882, 16148, 48952, 14259, 41110, 17308, 68873, 31616
Offset: 2

Views

Author

Eric M. Schmidt, Aug 29 2021

Keywords

Comments

Breuer's book erroneously gives a(33) = 18904. (See errata.)

References

  • Thomas Breuer, Characters and automorphism groups of compact Riemann surfaces, Cambridge University Press, 2000, p. 91.

Links

Crossrefs

Cf. A179982, A346293, A347368, A347369, A347370, A347371, A347372.
Showing 1-7 of 7 results.

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