A055085 Smallest integer containing all digits in all bases from 2 to n.

2, 11, 75, 978, 8350, 160773, 2217404, 45623244, 1093265784, 37206483195, 1587937206284, 109746121381518, 2697545063614180, 64810523515743579, 21538286780426129155

Offset: 2

Author

Ulrich Schimke (ulrschimke(AT)aol.com)

Keywords

• nonn
• base

Comments

a(12) is the first term whose expansion in base n does not begin with a 1. - Tom Womack

By definition, for n >= 3, a(n) > A000169(n), with equality when n=2 only. - Michel Marcus, Nov 10 2013

a(16) is the first term which is bigger than n^n, i.e., whose expansion in base n=16 is not a permutation of 16 different digits. - Ignat Soroko, Aug 14 2017

Examples

			75 is the smallest integer containing 0 and 1 in base 2 (1001011), 0, 1 and 2 in base 3 (2210) and 0, 1, 2 and 3 in base 4 (1023), hence a(4) = 75.
a(12) = 1587937206284 is 217904B5A638 in base 12; 562493178A90 in base 11.
		

Links

• Project Euler, Problem 571: Super Pandigital Numbers

Crossrefs

A051640 uses a weaker definition.

Programs

PARI

isok(i, n) = {for (b = 2, n, if (#Set(digits(i, b)) != b, return (0));); return (1);}
a(n) = {i = n^(n-1); while (! isok(i, n), i++); i;} \\ Michel Marcus, Nov 10 2013

Python

from itertools import count, product
from sympy.utilities.iterables import multiset_permutations
from gmpy2 import digits, mpz
def A055085(n): # assumes n <= 62
    dlist = tuple(digits(d,n) for d in range(n))
    for l in count(n-1):
        for t in product(dlist,repeat=l-n+1):
            for d in range(1,n):
                for u in multiset_permutations(sorted(t+dlist[:d]+dlist[d+1:])):
                    m = mpz(''.join((dlist[d],)+tuple(u)),n)
                    for b in range(n-1,1,-1):
                        if len(set(digits(m,b))) < b:
                            break
                    else:
                        return int(m) # Chai Wah Wu, Mar 14 2022

Extensions

a(11) from David Wasserman, Mar 25 2002

a(12) from Tom Womack (tom(AT)womack.net), Jun 19 2007

a(13)-a(15) from Ignat Soroko, Jan 05 2017

a(16) from Ignat Soroko, Aug 14 2017