The OEIS Foundation is supported by donations from users of the OEIS and by a grant from the Simons Foundation. Hints (Greetings from The On-Line Encyclopedia of Integer Sequences!)
 A263308 Smallest prime modulus p such that there exists a multiplicative-coset Ramsey algebra in n colors over Z/pZ, or 0 if no such prime exists. 4
 2, 5, 13, 41, 71, 97, 491, 0, 523, 1181, 947, 769, 0, 1709, 1291, 1217, 4013, 2521, 1901, 2801, 1933, 3257, 3221, 4129, 3701, 4889, 5563, 8849, 6323, 5521, 6263, 5441, 8779, 7481, 7841, 10009, 13469, 12161, 8971, 14561, 13367, 19993, 14621, 12497, 14401, 14537, 20117, 18913, 22541, 22901, 19687, 29537 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,1 COMMENTS a(8) = 0 means there is NO prime satisfying the condition. a(n) is known for 1 <= n <= 2000. LINKS Jeremy F. Alm, Table of n, a(n) for n = 1..2000 Jeremy F. Alm and Jacob Manske, Sum-free cyclic multi-bases and constructions of Ramsey algebras, Discrete Applied Mathematics, (180), Jan 10 2015, pp. 204-212. (arXiv:1307.0889 [math.CO], 2013-2014.) Jeremy F. Alm, 401 and beyond: improved bounds and algorithms for the Ramsey algebra search, Journal of Integer Sequences, Vol. 20 (2017), Article 17.8.4. (Also here: arXiv:1609.01817  [math.NT], 2016.) Tomasz Kowalski, Representability of Ramsey Relation Algebras, Algebra Universalis, Volume 74, Issue 3-4, November 2015, pp. 265-275. PROG (Python) import numpy as np import itertools from copy import copy from sympy.ntheory.residue_ntheory import primitive_root def psieve():     for n in [2, 3, 5, 7]:         yield n     D = {}     ps = psieve()     next(ps)     p = next(ps)     assert p == 3     psq = p*p     for i in itertools.count(9, 2):         if i in D:             step = D.pop(i)         elif i < psq:             yield i             continue         else:             assert i == psq             step = 2*p             p = next(ps)             psq = p*p         i += step         while i in D:             i += step         D[i] = step def check_p_m_v6(p, m, g):     ''' checks a prime p with primitive root g for m colors ''' X0 = np.array( [pow(g, i, p) for i in range(0, p-m, m) ] ) certificates = np.array([ pow(g, i, p) for i in range(m) ])     C_minus_X0 = ( ( certificates[:, np.newaxis] - X0 ) % p )     C_minus_X0_sets = [ set(L) for L in C_minus_X0 ] for i in range(m): Xi = {pow(g, x+i, p) for x in range(0, p-m, m)} for j in range(i, m):             if bool(Xi.intersection(C_minus_X0_sets[j])) == bool(j == 0):                 return False     return True def main(mikelist):     ''' Accepts a list of m's, checks all candidate primes until it finds one that works.         Will NOT terminate for m=8 or m=13 '''     lget = primitive_root     ### GIVE FUNCTIONS LOCAL NAMES ###     lcheck = check_p_m_v6     with open("401output.csv", 'a') as file:         for mike in mikelist:             primes = psieve()             prime = next(primes)             while prime < 2*mike**2 - 4*mike:                 prime = next(primes)             while True:                 if (prime-1)/2 % mike == 0:                     gen = lget(prime)                     p_out = lcheck(prime, mike, gen)                     if p_out == True:                         print mike, prime, gen                         file.write(str(mike) + ', ' + str(prime) + ', ' + str(gen) + '\n')                         break                 prime = next(primes) mrange = range(2, 8) + range(9, 13) + range(14, 101)  # a good place to start main(mrange) CROSSREFS Sequence in context: A149867 A062704 A274909 * A288388 A339224 A247981 Adjacent sequences:  A263305 A263306 A263307 * A263309 A263310 A263311 KEYWORD nonn AUTHOR Jeremy F. Alm, Oct 13 2015 EXTENSIONS More terms from Jeremy F. Alm, Sep 05 2016 STATUS approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent
The OEIS Community | Maintained by The OEIS Foundation Inc.

Last modified May 11 14:58 EDT 2021. Contains 343791 sequences. (Running on oeis4.)