A113028 a(n) is the largest integer whose base n digits are all different that is divisible by each of its individual digits.

1, 2, 54, 108, 152, 16200, 2042460, 4416720, 9867312, 2334364200, 421877610, 1779700673520, 4025593863720, 8605596007008, 1147797065081426760, 2851241701975626960, 6723295828605676320, 5463472083393768444000, 32677216797923569872, 29966837620559153371200

Offset: 2

Author

Peter Boothe, Jan 03 2006

Keywords

• nonn
• base

Comments

Note that this definition precludes the digit 0 appearing anywhere in the base n representation of the number.

Examples

			a(2) = 1 trivially because that is the only number in base 2 that does not contain 0.
a(4) = 54 because in base 4, 54 is 312_4. There is only one number containing different digits and no zeros higher than that, namely 321_4, but 321_4 is not divisible by 2.
		

References

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

Links

• Jes Wolfe, Table of n, a(n) for n = 2..48
• Michael S. Branicky, Python program
• Jes Wolfe, Even faster python program

Programs

Python

# see link for a faster version
from itertools import permutations
def fromdigits(d, b):
    n = 0
    for di in d: n *= b; n += di
    return n
def a(n):
    for digits in range(n-1, 0, -1):
        for p in permutations(range(n-1, 0, -1), r=digits):
            t = fromdigits(p, n)
            if all(t%di == 0 for di in p):
                return t
print([a(n) for n in range(2, 11)]) # Michael S. Branicky, Jan 17 2022

Extensions

a(11)-a(13) from Francis Carr (fcarr(AT)alum.mit.edu), Feb 08 2006

a(14) from Michael S. Branicky, Jan 17 2022

a(15)-a(17) from Michael S. Branicky, Jan 20 2022

a(18)-a(21) from Jes Wolfe, Apr 26 2024