James Hammer - cp's OEIS frontend

A317244 For n >= 3, smallest prime number N such that for every prime p >= N, every element in Z_p can be expressed as a sum of two n-gonal numbers mod p, without allowing zero as a summand.

11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 11, 13, 11, 11, 11, 11, 11, 11, 11, 23, 11, 11, 13, 29, 11, 11, 11, 11, 11, 11, 11, 37, 11, 13, 11, 11, 11, 11, 23, 11, 11, 11, 11, 47, 13, 11, 29, 53, 11, 11, 11, 11, 11, 11, 11, 13, 11, 23, 11, 61, 11, 11, 37, 11, 11, 11, 13, 71, 11, 29, 11, 73, 11, 11, 11, 11, 23, 13, 11, 83, 11, 11, 11, 89, 11, 11, 47, 11, 13, 11, 11, 11, 29, 37, 53, 23, 11

Offset: 3

Theresa Baren, James Hammer, Joshua Harrington, Ziyu Liu, Sean E. Rainville, Melea Roman, Hongkwon V. Yi, Jul 24 2018

nonn

Joshua Harrington, Lenny Jones, and Alicia Lamarche, Representing Integers as the Sum of Two Squares in the Ring Z_n, Journal of Integer Sequences, Vol. 17 (2014), Article 14.7.4.
Bernard M. Moore and H. Joseph Straight, Pythagorean triples in multiplicative groups of prime power order, Pi Mu Epsilon Journal, vol. 14, no. 3, 2015, pp. 191-198.

A320578 Triangle read by rows: T(n,k) is the number of permutation graphs on n vertices with domination number k, with 1 <= k <= n.

Original entry on oeis.org

1, 1, 1, 3, 2, 1, 10, 10, 3, 1, 43, 54, 18, 4, 1, 223, 351, 113, 27, 5, 1, 1364, 2613, 833, 186, 37, 6, 1, 9643, 21965, 6921, 1461, 274, 48, 7, 1, 77545, 205780, 64128, 12727, 2253, 378, 60, 8, 1, 699954, 2127068, 655391, 122345, 20230, 3230, 499, 73, 9, 1

Offset: 1

James Hammer, Daniel A. McGinnis, Oct 15 2018

			Triangle begins:
    1;
    1,   1;
    3,   2,   1;
   10,  10,   3,   1;
   43,  54,  18,   4,   1;
  223, 351, 113,  27,   5,   1;
  ...

Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.

Cf. A320579, A320583

import networkx as nx
import math
def permutation(lst):
    if len(lst) == 0:
        return []
    if len(lst) == 1:
        return [lst]
    l = []
    for i in range(len(lst)):
        m = lst[i]
        remLst = lst[:i] + lst[i + 1:]
        for p in permutation(remLst):
            l.append([m] + p)
    return l
def generatePermsOfSizeN(n):
    lst = []
    for i in range(n):
        lst.append(i+1)
    return permutation(lst)
def powersetHelper(A):
    if A == []:
        return [[]]
    a = A[0]
    incomplete_pset = powersetHelper(A[1:])
    rest = []
    for set in incomplete_pset:
        rest.append([a] + set)
    return rest + incomplete_pset
def powerset(A):
    ps = powersetHelper(A)
    ps.sort(key = len)
    return ps
    print(ps)
def countDomNumbersOnN(n):
    lst=[]
    perms = generatePermsOfSizeN(n)
    for i in range(n):
        lst.append(i+1)
    ps = powerset(lst)
    dic={}
    for perm in perms:
        tempGraph = nx.Graph()
        tempGraph.add_nodes_from(perm)
        for i in range(len(perm)):
            for k in range(i+1, len(perm)):
                if perm[k] < perm[i]:
                    tempGraph.add_edge(perm[i], perm[k])
        for p in ps:
            if nx.is_dominating_set(tempGraph,p):
                dom = len(p)
                if dom in dic:
                    dic[dom] += 1
                    break
                else:
                    dic[dom] = 1
                    break
    return dic

T(n,k) = A320579(n,k) + A320583(n,k).

T(n,1) = A320583(n,1).

A320579 Triangle read by rows: T(n,k) is the number of disconnected permutation graphs on n vertices with domination number k, with 2 <= k <= n.

Original entry on oeis.org

1, 2, 1, 7, 3, 1, 26, 18, 4, 1, 115, 111, 27, 5, 1, 592, 771, 186, 37, 6, 1, 3532, 5906, 1459, 274, 48, 7, 1, 24212, 49982, 12643, 2253, 378, 60, 8, 1, 188869, 466314, 120252, 20228, 3230, 499, 73, 9, 1

Offset: 2

James Hammer, Daniel A. McGinnis, Oct 15 2018

			Triangle begins:
    1;
    2,   1;
    7,   3,   1;
   26,  18,   4,  1;
  115, 111,  27,  5,  1;
  592, 771, 186, 37,  6,  1;
  ...

Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.

CF. A320578, A320583.

import networkx as nx
import math
def permutation(lst):
    if len(lst) == 0:
        return []
    if len(lst) == 1:
        return [lst]
    l = []
    for i in range(len(lst)):
        m = lst[i]
        remLst = lst[:i] + lst[i + 1:]
        for p in permutation(remLst):
            l.append([m] + p)
    return l
def generatePermsOfSizeN(n):
    lst = []
    for i in range(n):
        lst.append(i+1)
    return permutation(lst)
def powersetHelper(A):
    if A == []:
        return [[]]
    a = A[0]
    incomplete_pset = powersetHelper(A[1:])
    rest = []
    for set in incomplete_pset:
        rest.append([a] + set)
    return rest + incomplete_pset
def powerset(A):
    ps = powersetHelper(A)
    ps.sort(key = len)
    return ps
    print(ps)
def countdisDomNumbersOnN(n):
    lst=[]
    l=[]
    perms = generatePermsOfSizeN(n)
    for i in range(n):
        lst.append(i+1)
    ps = powerset(lst)
    dic={}
    for perm in perms:
        tempGraph = nx.Graph()
        tempGraph.add_nodes_from(perm)
        for i in range(len(perm)):
            for k in range(i+1, len(perm)):
                if perm[k] < perm[i]:
                    tempGraph.add_edge(perm[i], perm[k])
        if nx.is_connected(tempGraph)==False:
            for p in ps:
                if nx.is_dominating_set(tempGraph,p):
                    dom = len(p)
                    if dom in dic:
                        dic[dom] += 1
                        break
                    else:
                        dic[dom] = 1
                        break
    return dic

T(n,k) = A320578(n,k) - A320583(n,k).

A320583 Irregular triangle read by rows: T(n,k) is the number of connected permutation graphs on n vertices with domination number k, with 1 <= k <= floor(n/2).

Original entry on oeis.org

1, 1, 3, 10, 3, 43, 28, 223, 236, 2, 1364, 1842, 62, 9643, 18433, 1015, 2, 77545, 181568, 14146, 84, 699954, 1938199, 189077, 2093, 2

Offset: 1

James Hammer, Daniel A. McGinnis, Oct 15 2018

			Triangle begins:
    1;
    1;
    3;
   10,   3;
   43,  28;
  223, 236,  2;
  ...

Theresa Baren, Michael Cory, Mia Friedberg, Peter Gardner, James Hammer, Joshua Harrington, Daniel McGinnis, Riley Waechter, Tony W. H. Wong, On the Domination Number of Permutation Graphs and an Application to Strong Fixed Points, arXiv:1810.03409 [math.CO], 2018.

Cf. A320578, A320579.

import networkx as nx
import math
def permutation(lst):
    if len(lst) == 0:
        return []
    if len(lst) == 1:
        return [lst]
    l = []
    for i in range(len(lst)):
        m = lst[i]
        remLst = lst[:i] + lst[i + 1:]
        for p in permutation(remLst):
            l.append([m] + p)
    return l
def generatePermsOfSizeN(n):
    lst = []
    for i in range(n):
        lst.append(i+1)
    return permutation(lst)
def powersetHelper(A):
    if A == []:
        return [[]]
    a = A[0]
    incomplete_pset = powersetHelper(A[1:])
    rest = []
    for set in incomplete_pset:
        rest.append([a] + set)
    return rest + incomplete_pset
def powerset(A):
    ps = powersetHelper(A)
    ps.sort(key = len)
    return ps
    print(ps)
def countcnctdDomNumbersOnN(n):
    lst=[]
    l=[]
    perms = generatePermsOfSizeN(n)
    for i in range(n):
        lst.append(i+1)
    ps = powerset(lst)
    dic={}
    for perm in perms:
        tempGraph = nx.Graph()
        tempGraph.add_nodes_from(perm)
        for i in range(len(perm)):
            for k in range(i+1, len(perm)):
                if perm[k] < perm[i]:
                    tempGraph.add_edge(perm[i], perm[k])
        if nx.is_connected(tempGraph)==True:
            for p in ps:
                if nx.is_dominating_set(tempGraph,p):
                    dom = len(p)
                    if dom in dic:
                        dic[dom] += 1
                        break
                    else:
                        dic[dom] = 1
                        break
    return dic

T(n,n/2) = 2 for even n. See Theorem 4.5 in the link by Theresa Baren, et al.
T(n,k) = A320578(n,k) - A320579(n,k).
T(n,1) = A320578(n,1).

A289849 Cardinality of the maximal set of ordered factor pairs such that a Quasi-Factor Pair Latin Square of order n exists.

Original entry on oeis.org

1, 2, 2, 3, 2, 4, 2, 4, 3, 4, 2, 5, 2, 4, 4, 5, 2, 6, 2, 5, 4, 4, 2, 6, 3, 4, 4, 5, 2

Offset: 1

Leah S. Miller, Jordan Lenchitz, Makkah Davis, Bob Kuo, Boyang Su, James Hammer, Jul 13 2017

For n a prime power or twice a prime, a(n) is known to coincide with A000005(n).

James Hammer, Factor Pair Latin Squares, Dissertation, Auburn University, 2015.

Cf. A000005.

A289812 n for which a Factor Pair Latin Square of order n exists.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 18, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32

Offset: 1

Jordan Lenchitz, Leah S. Miller, Makkah Davis, Bob Kuo, Boyang Su, James Hammer, Jul 12 2017

This sequences differs from A206551 because no Factor Pair Latin Square of order 45 exists.

It is known that every prime power (A000961) as well as twice every prime number (A100484) appears in this sequence.

James Hammer, Factor Pair Latin Squares, Dissertation, Auburn University, 2015.

Cf. A000961, A100484, A206551.

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User: James Hammer

A317244 For n >= 3, smallest prime number N such that for every prime p >= N, every element in Z_p can be expressed as a sum of two n-gonal numbers mod p, without allowing zero as a summand.

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A320578 Triangle read by rows: T(n,k) is the number of permutation graphs on n vertices with domination number k, with 1 <= k <= n.

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A320579 Triangle read by rows: T(n,k) is the number of disconnected permutation graphs on n vertices with domination number k, with 2 <= k <= n.

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A320583 Irregular triangle read by rows: T(n,k) is the number of connected permutation graphs on n vertices with domination number k, with 1 <= k <= floor(n/2).

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Examples

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Programs

Python

Formula

A289849 Cardinality of the maximal set of ordered factor pairs such that a Quasi-Factor Pair Latin Square of order n exists.

Views

Author

Keywords

Comments

Links

Crossrefs

A289812 n for which a Factor Pair Latin Square of order n exists.

Views

Author

Keywords

Comments

Links

Crossrefs