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

1,2

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]

Prog

(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

Crossrefs

Cf. A169912, A067139.

Base 3 lunar primes: A130206, A170806.

Sequence in context: A305379 A004685 A361335 * A346821 A222365 A041216

Adjacent sequences: A170997 A170998 A170999 * A171001 A171002 A171003

Keyword

nonn

Author

N. J. A. Sloane, Aug 31 2010

Status

approved