A384639 -id:A384639 - cp's OEIS frontend

A387066 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

1, 4, 21, 191, 3338, 115438

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the least separating genus.

			For genus 0: only the circle.
For genus 1: 2 circles, bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.
J. Bernhard and J. J. P. Veerman, The Topology of Surface Mediatrices, Topology and its Applications, 154, 54-68, 2007.

Cf. A384639, A387067, A387068.

A387067 Number of equivalence classes (up to graph homeomorphism) of finite,connected graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

Original entry on oeis.org

1, 3, 17, 164, 3096, 111445

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the minimal separating genus.

			For genus 0: only the circle.
For genus 1: bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.

Cf. A384639, A387066, A387068.

A387068 Number of equivalence classes (up to homeomorphism) of finite, connected ribbon graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

Original entry on oeis.org

1, 3, 31, 1831, 462638, 243066565

Offset: 0

J. J. P. Veerman, Aug 15 2025

n is called the least separating genus.

These numbers are larger than those of A387067, because different embeddings of the same graph are counted separately. For instance, there is a ribbon graph for the bouquet of 3 circles with least separating genus 1 and a different embedding with least separating genus 2.

			For genus 0: thickened circle.
For genus 1: thickened versions of a bouquet of 2 circles, bouquet of 3 circles, 4-fold multi-edge.

C. N. Aagaard and J. J. P. Veerman, Classification of Minimal Separating Sets of Low Genus Surfaces, Topology and its Applications, Accepted, 2025.

Cf. A384639, A387066, A387067.

A384835 The exponents (j, k) of the numbers 2^j*3^k that are averages of twin primes, with both j and k > 0, in the order of their sum, and then by j.

Original entry on oeis.org

1, 1, 1, 2, 2, 1, 2, 3, 3, 2, 4, 3, 6, 1, 5, 4, 7, 2, 3, 10, 6, 7, 2, 15, 12, 5, 18, 1, 18, 5, 21, 4, 24, 5, 27, 4, 11, 24, 30, 7, 32, 9, 33, 8, 31, 12, 36, 7, 43, 2, 32, 15, 43, 8, 50, 9, 63, 2, 66, 25, 79, 20, 99, 10, 57, 64, 82, 63, 63, 88, 56, 99, 148, 27

Offset: 1

Ken Clements, Jun 10 2025

These are the (j,k) exponents of the numbers 2^j*3^k that are averages of twin primes, ordered by j+k, j. They are remarkable in structure because except for the first pair (2^1*3^1), j+k is always an odd number. I have proof of this, and the reason it is not the case for the first pair is that 6-1=5 is the only number divisible by 5 that is prime.

			2^a(1) * 3^a(2) = 6.
2^a(3) * 3^a(4) = 18.
2^a(5) * 3^a(6) = 12.
2^a(7) * 3^a(8) = 108.
2^a(9) * 3^a(10) = 72.

Ken Clements, Table of n, a(n) for n = 1..162

Cf. A027856, A384639 (ordered by value of 2^j*3^k).

seq[max_] := Flatten@ Transpose[IntegerExponent[Select[Flatten[Table[2^j*3^(m-j), {m, 2, max}, {j, 1, m-1}]], And @@ PrimeQ[# + {-1, 1}] &], #] & /@ {2, 3}]; seq[200] (* Amiram Eldar, Jun 26 2025 *)

from sympy import isprime
def is_TP_pi_2(j, k):
    N = 2**j * 3**k
    return isprime(N-1) and isprime(N+1)
def aupto(limit):
    result = [1, 1]
    for exponent_sum in range(3, limit+1, 2):
        for j in range(1, exponent_sum):
             k = exponent_sum - j
             if is_TP_pi_2(j, k):
                  result.append(j)
                  result.append(k)
    return result
print(aupto(10_000))

import heapq
from gmpy2 import is_prime
from itertools import islice
def agen(): # generator of terms
    v, oldv, h = 1, 0, [(2, 1, 1, 6)]
    while True:
        s, e2, e3, v = heapq.heappop(h)
        if v != oldv:
            if is_prime(v-1) and is_prime(v+1):
                yield from (e2, e3)
            oldv = v
            heapq.heappush(h, (s+1, e2+1, e3, 2*v))
            heapq.heappush(h, (s+1, e2, e3+1, 3*v))
print(list(islice(agen(), 70))) # Michael S. Branicky, Jun 26 2025

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A387066 Number of equivalence classes (up to graph homeomorphism) of finite graphs that have an embedding in an orientable surface of genus n which minimally separates the surface of genus n (that is, no proper subset of the embedding separates the genus n surface) but not the surface of genus n-1.

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A384835 The exponents (j, k) of the numbers 2^j*3^k that are averages of twin primes, with both j and k > 0, in the order of their sum, and then by j.

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