A130206 Primes in lunar arithmetic in base 3 written in base 10.

5, 6, 7, 8, 11, 19, 20, 23, 29, 32, 34, 35, 38, 46, 47, 55, 56, 58, 59, 61, 62, 64, 65, 68, 71, 73, 74, 77, 83, 86, 88, 89, 95, 97, 98, 103, 104, 106, 107, 110, 119, 127, 128, 136, 137, 142, 143, 145, 146, 154, 155, 163, 164, 166, 167, 169, 170, 173, 175, 176, 178, 179, 184, 185, 187, 188, 190

Offset: 1

Author

N. J. A. Sloane, Sep 30 2010

Keywords

• nonn
• base

Links

• 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]
• Index entries for sequences related to dismal (or lunar) arithmetic

Crossrefs

Cf. A170806.

Programs

Python

def ternary(m):
    if m == 0: return '0'
    s = []
    while m:
        m, r = divmod(m, 3)
        s.append(str(r))
    return ''.join(reversed(s))
def addn(m1, m2):
    s1, s2 = ternary(m1), ternary(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 =  ternary(m1), ternary(m2), '0'
    for i in range(len(s2)):
        k = s2[-i-1]
        prod = addn(int(str(prod), 3), int(''.join(min(j, k) for j in s1), 3)*3**i)
    return prod
for m in range(3,201):
    i, ct = 1, 0
    for i in range(1, m+1):
        if i == 2: continue
        j = i
        for j in range(1, m+1):
            if j == 2: continue
            ij = int(str(muln(i, j)), 3)
            if ij == m: ct += 1; break
        if ct > 0: break
if ct == 0: print(m) # Ya-Ping Lu, Dec 30 2020

Extensions

Entries >=83 from R. J. Mathar, Nov 23 2015