A171000 Irreducible Boolean polynomials written as binary vectors.

1, 10, 11, 101, 1001, 1011, 1101, 10001, 10011, 10111, 11001, 11101, 100001, 100011, 100101, 100111, 101001, 101011, 110001, 110101, 111001, 1000001, 1000011, 1000101, 1000111, 1001011, 1001101, 1001111, 1010001, 1010011, 1010111, 1011001, 1011101, 1100001

Offset: 1

Author

N. J. A. Sloane, Aug 31 2010

Keywords

• nonn

Comments

These are the polynomials enumerated in A169912, and written in base 10 in A067139.

This sequence consists of 1 and the lunar primes in base 2 arithmetic. To construct the lunar base 2 primes, start with 10, and repeatedly adjoin the next smallest binary number that is not a lunar base-2 multiple of any earlier number. - N. J. A. Sloane, Jan 26 2011

Links

• N. J. A. Sloane, Table of n, a(n) for n = 1..5655
• D. Applegate, M. LeBrun and N. J. A. Sloane, Dismal Arithmetic, arXiv:1107.1130 [math.NT], 2011. [Note: we have now changed the name from "dismal arithmetic" to "lunar arithmetic" - the old name was too depressing]

Crossrefs

Cf. A169912, A067139.

Base 3 lunar primes: A130206, A170806.

Programs

Python

def addn(m1, m2):
    s1, s2 = "{0:b}".format(m1), "{0:b}".format(m2)
    len_max = max(len(s1), len(s2))
    return int(''.join(max(i, j) for i, j in zip(s1.rjust(len_max, '0'), s2.rjust(len_max, '0'))))
def muln(m1, m2):
    s1, s2, prod =  "{0:b}".format(m1), "{0:b}".format(m2), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod), 2), int(''.join(min(j, k) for j in s1), 2)*2**i)
    return prod
L_p10, m = [1], 2
while m < 100:
    ct = 0
    for i in range(1, len(L_p10)):
        p = L_p10[i]
        for j in range(2, m):
            jp = int(str(muln(j, p)), 2)
            if jp > m: break
            if jp == m: ct += 1; break
        if ct > 0: break
    if ct == 0: L_p10.append(m)
    m += 1
L_p2 = []
for d in L_p10: L_p2.append("{0:b}".format(d))
print(*L_p2, sep =', ') # Ya-Ping Lu, Dec 27 2020