A114342 Largest number whose base-n representation does not contain any digit more than once and which is not divisible by any of its base-n digits, or 0 if no such number exists.

0, 0, 0, 35, 577, 5909, 16331, 2053379, 42374099, 987654203, 2334368201, 736867783013, 23136292864661, 789018236128979, 1936265501684027, 1147797409030816259, 48471109094902544503, 2178347851919531380901, 5463472083532379956913, 5228356786703601108032803

Offset: 1

Author

Francis Carr (fcarr(AT)alum.mit.edu), Feb 08 2006

Keywords

• nonn
• base

Examples

			There are 49 numbers whose base-4 representation does not contain repeated digits. Of these, the largest which is not divisible by any of its digits is a(4) = 203_4 = 35_10.
Any base-3 number containing only 0's and 2's with at least one 2 is divisible by 2, while any number with a 1 is divisible by 1, so no positive integer meets the criteria in base 3. Thus a(3) = 0.
		

References

• "Enigma 1343: Digital Dividend", New Scientist, Jun 04 2005, 28.

Links

• Enigmatic Code, Enigma 1343: Digital Dividend, from New Scientist, Jun 04 2005, 28.

Crossrefs

Cf. A113028.

Programs

Sage

def A114342(n):
    dd = [0] + [2..n-1]
    for width in [1..n-1][::-1]:
        found = []
        for dc in Combinations(dd, width):
            m = sum(dc) % (n-1)
            if gcd(m,n-1) in dc: continue # rule of nines
            for p in Permutations(dc[::-1]):
                s = sum((d)*n**i for i,d in enumerate(p[::-1]))
                if not any(d != 0 and s % d == 0 for d in p): found.append(s)
                if found and width == len(dd): return s
                if found and s < max(found): break
        if found: return max(found)
    return 0 # D. S. McNeil, Oct 01 2011

Extensions

a(12)-a(20) from Nathaniel Johnston, Sep 30 2011