Eric M. Schmidt - cp's OEIS frontend

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

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

Eric M. Schmidt, Aug 29 2021

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

Breuer's book erroneously gives a(33) = 949. (See errata.)

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

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

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

Eric M. Schmidt, Aug 29 2021

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

Breuer's book erroneously gives a(33) = 576. (See errata.)

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

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

Eric M. Schmidt, Aug 29 2021

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

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

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

Eric M. Schmidt, Aug 29 2021

This includes subgroups of the full automorphism group.

Breuer's book erroneously gives a(33) = 1013. (See errata.)

			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]

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

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

Eric M. Schmidt, Aug 29 2021

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

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

Cf. A179982, A346293, A347368, A347369, A347370, A347371, A347372.

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

Eric M. Schmidt, Aug 29 2021

This includes subgroups of the full automorphism group.

Breuer's book erroneously gives a(33) = 2843. (See errata.)

			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.

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

Thomas Breuer, Errata et Addenda for Characters and Automorphism Groups of Compact Riemann Surfaces.
Jen Paulhus, Branching data for curves up to genus 48.

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

A343821 Numbers k such that the alternating group A_k is a Hurwitz group.

Original entry on oeis.org

15, 21, 22, 28, 29, 35, 36, 37, 42, 43, 45, 49, 50, 51, 52, 56, 57, 58, 63, 64, 65, 66, 70, 71, 72, 73, 77, 78, 79, 80, 81, 84, 85, 86, 87, 88, 91, 92, 93, 94, 96, 98, 99, 100, 101, 102, 105, 106, 107, 108, 109, 112, 113, 114, 115, 116, 117, 119, 120, 121, 122

Offset: 1

Eric M. Schmidt, Apr 30 2021

nonn

This sequence contains every k > 167 [Conder].

This sequence is found in Section 2 of the Gordejuela and Martínez paper, which has a slight error: 86 occurs twice and 87 is missing.

Eric M. Schmidt, Table of n, a(n) for n = 1..150
Marston D. E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), pp. 75-86.
J. J. Etayo Gordejuela and E. Martínez, Alternating groups, Hurwitz groups and H^*-groups, Journal of Algebra, 283 (2005), pp. 327-349.

Complement of A343822.

Cf. A179982.

A343822 Numbers k such that the alternating group A_k is not a Hurwitz group.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 16, 17, 18, 19, 20, 23, 24, 25, 26, 27, 30, 31, 32, 33, 34, 38, 39, 40, 41, 44, 46, 47, 48, 53, 54, 55, 59, 60, 61, 62, 67, 68, 69, 74, 75, 76, 82, 83, 89, 90, 95, 97, 103, 104, 110, 111, 118, 125, 131, 139, 146, 167

Offset: 1

Eric M. Schmidt, Apr 30 2021

Complement of A343821.

See the classification in section 5 of the first Conder reference. The term 139 is erroneously omitted there, as pointed out in the second Conder reference [Section 3].

Marston D. E. Conder, Generators for alternating and symmetric groups, J. London Math. Soc. (2) 22 (1980), pp. 75-86.
Marston D. E. Conder, Some results on quotients of triangle groups, Bulletin of the Australian Mathematical Society, Volume 30, Issue 1, August 1984, pp. 73-90.
J. J. Etayo Gordejuela and E. Martínez, Alternating groups, Hurwitz groups and H^*-groups, Journal of Algebra, 283 (2005), pp. 327-349.

Cf. A179982, A343821.

A342824 Number of ways n appears as a cross-polytope number (A142978).

Original entry on oeis.org

1, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 2, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 1, 2

Offset: 2

Eric M. Schmidt, Mar 22 2021

nonn

Every entry in the first column (of A142978) is 1, so this sequence starts at a(2).
a(n) is always positive, as the first row lists the positive integers.
a(n) >= 3 infinitely often. This happens, in particular, at every even square > 4. (The second row contains the squares, and the second column the positive even numbers.)
For n <= 10000, the only instance of a(n) > 3 is a(1156) = 4. This occurs because 1156 is even, square, and octahedral (third row of A142978).

Cf. A142978.

def a(n) : return len([K for K in [2..n] if n == next(A142978(N, K) for N in (1..) if A142978(N, K) >= n)])

A296613 Smallest k such that either k >= n and k is a power of 2, or k >= 5n/3 and the prime divisors of k are precisely 2 and 5.

Original entry on oeis.org

1, 2, 4, 4, 8, 8, 8, 8, 16, 16, 16, 16, 16, 16, 16, 16, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 32, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 64, 128, 128

Offset: 1

Eric M. Schmidt, Dec 16 2017

nonn

First disagreement with A062383(n-1) is at n = 129.

For n > 2, a(n) is not squarefree. - Iain Fox, Dec 17 2017

Eric M. Schmidt, Table of n, a(n) for n = 1..10000
Bernadette Faye, Florian Luca, Pieter Moree, On the discriminator of Lucas sequences, arXiv:1708.03563 [math.NT], 2017, Theorem 1.

Cf. A033846.

a(n) = for(k=n, +oo, if(k == 2^valuation(k, 2) || (k >= 5*n/3 && factor(k)[, 1] == [2, 5]~), return(k))) \\ Iain Fox, Dec 17 2017

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User: Eric M. Schmidt

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

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

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

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

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

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

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A343821 Numbers k such that the alternating group A_k is a Hurwitz group.

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A343822 Numbers k such that the alternating group A_k is not a Hurwitz group.

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A342824 Number of ways n appears as a cross-polytope number (A142978).

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A296613 Smallest k such that either k >= n and k is a power of 2, or k >= 5n/3 and the prime divisors of k are precisely 2 and 5.

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