Alexei Vernitski - cp's OEIS frontend

A380650 The largest number which is a linear combination of the divisors of n with nonnegative integer coefficients such that no linear combination with smaller nonnegative integer coefficients is equal to n.

0, 1, 2, 3, 4, 7, 6, 7, 8, 13, 10, 17, 12, 19, 22, 15, 16, 25, 18, 31, 32, 31, 22, 37, 24, 37, 26, 45, 28, 60, 30, 31, 52, 49, 58, 59, 36, 55, 62, 67, 40, 85, 42, 73, 76, 67, 46, 77, 48, 73, 82, 87, 52, 79, 94, 97, 92, 85, 58

Offset: 1

Alexei Vernitski, Jan 29 2025

nonn

The mean of this sequence and Euler's totient function A000010 is approximately (but not exactly) equal to n.
The definition has evolved from a recreational question asked by P. M. Higgins, asking what maximal sum of money can be produced using British coins so no sum of one pound is produced by any subset of these coins.
The terms up to and including a(29)=28 agree with the formula a(n) = (A145388(n) - 1)/2, but a(30)=60, while the formula gives 67. This difference should be confirmed by an independent calculation using the definition in the name. - Hugo Pfoertner, Feb 14 2025

			For n = 12, the largest sum is 17 = 0*1 + 0*2 + 1*3 + 2*4 + 1*6 = 0*1 + 0*2 + 3*3 + 2*4 + 0*6.
For n = 30, the largest sum is 60 = 1*1 + 0*2 + 0*3 + 0*5 + 4*6 + 2*10 + 1*15.

Cf. A000010, A145388.

A358653 a(n) is the number of trivial braids on 3 strands which are products of n generators a, b, where a = sigma_1 sigma_2 sigma_1 and b = sigma_1 sigma_2.

Original entry on oeis.org

1, 0, 4, 0, 28, 10, 244, 210, 2412, 3366, 26014, 49456, 299452, 701818, 3624478

Offset: 0

Alexei Vernitski, Nov 25 2022

In the discussion of A354602, Andrey Zabolotskiy asked what the values of the sequence would be if expressed in terms of a and b. This sequence lists these values.

Wikipedia, Relation with symmetric group and the pure braid group.

Cf. A354602.

B. = BraidGroup(3)
gen = [s1*s2*s1, s1*s2]
gen += [x^-1 for x in gen]
e = B(())
words, a = {e: 1}, [1]
for n in range(15):
    old_words, words = words, {}
    for w, c in old_words.items():
        for g in gen:
            nw = w*g
            words[nw] = words.get(nw, 0) + c
    a.append(words.get(e, 0))
print(a) # Andrey Zabolotskiy, Jan 16 2024

a(11)-a(14) from Andrey Zabolotskiy, Jan 16 2024

A354602 a(n) is the number of trivial braids on 3 strands with 2*n crossings.

Original entry on oeis.org

1, 4, 28, 244, 2412, 25804, 290932, 3403404, 40914508

Offset: 0

Alexei Vernitski, Jul 08 2022

In other words, a(n) is the number of products of 2*n generators in the braid group B_3 which are equal to the identity element of the group.
Only braids with an even number of crossings are considered because a braid with an odd number of crossings cannot be trivial.
If we do include the 0s corresponding to the odd values of the number of crossings, a group-theoretical name for this sequence is the cogrowth sequence of B_3.

Cf. A000984 (number of trivial braids on 2 strands with 2*n crossings), A047849 (number of trivial permutations of 3 elements after 2*n adjacent transpositions).

A338660 Number of circle graphs of Gauss diagrams of meander curves with 2n+1 crossings.

Original entry on oeis.org

1, 2, 5, 13, 43, 167

Offset: 1

Alexei Vernitski, Apr 22 2021

See A343358 for a definition of a graph corresponding to a closed planar curve. Meanders have been defined in various ways; for the purpose of considering their Gauss diagrams and graphs, a meander is understood as a closed planar curve in whose graph there is a vertex adjacent to every other vertex. This sequence is the number of distinct graphs of meanders of (necessarily odd) sizes.

Delecroix, Vincent, et al. "Enumeration of meanders and Masur-Veech volumes." Forum of Mathematics, Pi. Vol. 8. Cambridge University Press, 2020.
Grinblat, Andrey, and Viktor Lopatkin. "On realizabilty of Gauss diagrams and constructions of meanders." Journal of Knot Theory and Its Ramifications 29.05 (2020): 2050031.

V. Delecroix et al. Enumeration of meanders and Masur-Veech volumes, arXiv preprint arXiv:1705.05190 [math.GT], 2017-2019.
Andrey Grinblat and Viktor Lopatkin. On realizabilty of Gauss diagrams and constructions of meanders. arXiv:1808.08542 [math.AT], 2018.
Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin, and Alexei Vernitski, Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration, arXiv:2108.02873 [math.GT], 2021.

Cf. A343358.

A343358 Number of connected graphs with n vertices which are realizable (in the sense of realizability of Gauss diagrams).

Original entry on oeis.org

1, 1, 2, 3, 7, 18, 41, 123, 361, 1257, 4573

Offset: 3

Alexei Vernitski, Apr 12 2021

Consider a closed planar curve which crosses itself n times. Build a graph in which crossings are vertices, and two crossings c, d are not connected [connected] if respectively it is [is not] possible to travel along the curve from c to c without passing through d. A graph which can be produced in this way is called realizable. A classical related concept is that of a Gauss diagram (of a closed planar curve); realizable graphs are exactly the circle graphs of realizable Gauss diagrams.

The entries are produced by our code, and the entry for n=11 is corroborated by Section 4 in Bishler et al. which lists 6 pairs of alternating mutant knots of size 11. The entries for n=12, 13 are similarly corroborated by Stoimenow's data.

L. Bishler et al. "Distinguishing mutant knots." Journal of Geometry and Physics 159 (2021): 103928.

L. Bishler, et al., Distinguishing mutant knots, arXiv:2007.12532 [hep-th], 2021.
Abdullah Khan, Alexei Lisitsa, Viktor Lopatkin, and Alexei Vernitski, Circle graphs (chord interlacement graphs) of Gauss diagrams: Descriptions of realizable Gauss diagrams, algorithms, enumeration, arXiv:2108.02873 [math.GT], 2021.
Alexei Lisitsa, Abdullah Khan, and Alexei Vernitski, An experimental approach to Gauss diagram realizability, 28th British Comb. Conf., Durham Univ. (UK, 2021), p. 107.
Alexei Lisitsa and Alexei Vernitski, Counting graphs induced by Gauss diagrams and families of mutant alternating knots, Examples Counterex. (2024) Vol. 6, Art. No. 100162.
A. Stoimenow, Knot data tables.

Cf. A002864, which starts with 1, 1, 2, 3, 7, 18, 41, 123, 367. This is because an alternating prime knot with 10 or fewer crossings is uniquely defined by the graph of the corresponding closed planar curve. Only starting from n=11 some alternating knots which share the same graph but are distinct knots (called "mutant knots") start appearing.

Cf. A264759, which starts with 1, 1, 2, 3, 10; there is a mismatch starting from size 7. Indeed, starting from n=7 there are some planar curves which share the same graph but have distinct Gauss diagrams.

cp's OEIS Frontend

User: Alexei Vernitski

A380650 The largest number which is a linear combination of the divisors of n with nonnegative integer coefficients such that no linear combination with smaller nonnegative integer coefficients is equal to n.

Views

Author

Keywords

Comments

Examples

Crossrefs

A358653 a(n) is the number of trivial braids on 3 strands which are products of n generators a, b, where a = sigma_1 sigma_2 sigma_1 and b = sigma_1 sigma_2.

Views

Author

Keywords

Comments

Links

Crossrefs

Programs

SageMath

Extensions

A354602 a(n) is the number of trivial braids on 3 strands with 2*n crossings.

Views

Author

Keywords

Comments

Crossrefs

A338660 Number of circle graphs of Gauss diagrams of meander curves with 2n+1 crossings.

Views

Author

Keywords

Comments

References

Links

Crossrefs

A343358 Number of connected graphs with n vertices which are realizable (in the sense of realizability of Gauss diagrams).

Views

Author

Keywords

Comments

References

Links

Crossrefs