A090493 Least k such that n^k contains all the digits from 0 through 9, or 0 if no such k exists.

0, 68, 39, 34, 19, 20, 18, 28, 24, 0, 23, 22, 22, 21, 12, 17, 14, 21, 17, 51, 17, 18, 14, 19, 11, 18, 13, 11, 12, 39, 11, 14, 16, 14, 19, 10, 13, 14, 17, 34, 11, 17, 13, 16, 15, 11, 12, 12, 9, 18, 16, 11, 13, 10, 12, 7, 13, 11, 11, 20, 14, 18, 13, 14, 10, 13, 10, 9, 11, 18, 15

Offset: 1

Author

Amarnath Murthy, Dec 03 2003

Keywords

• base
• nonn

Comments

Note that the values of n for which a(n) = 1 have density 1.

Is it known that a(n)=0 only for n a power of 10? - Christopher J. Smyth, Aug 21 2014

a(n) >= ceiling(log_n(10)*9), whenever a(n)>0. This is because in order for an integer to have 10 digits its base-10 magnitude must be at least 9. - Ely Golden, Sep 06 2017

Examples

			a(5)=19: 5^19 = 19073486328125.
		

Links

• Seiichi Manyama, Table of n, a(n) for n = 1..10000 (terms 1..1000 from T. D. Noe)
• Ely Golden, Python program to generate a(n)

Crossrefs

Cf. A020665, A062518, A171102.

Exponents of powers of k that contain all ten decimal digits: A130694 (k=2), A236673 (k=3), A284670 (k=5), A284672 (k=7).

Programs

Maple

a:= proc(n) local k;
   if n = 10^ilog10(n) then return 0 fi;
   for k from 1 do
     if nops(convert(convert(n^k,base,10),set))=10 then return k fi
   od
end proc:
seq(a(n),n=1..100); # Robert Israel, Aug 20 2014

Mathematica

Table[If[IntegerQ@ Log10[n], 0, SelectFirst[Range[#, # + 100] &@ Ceiling[9 Log[n, 10]], NoneTrue[DigitCount[n^#], # == 0 &] &]], {n, 71}] (* Michael De Vlieger, Sep 06 2017 *)

PARI

a(n) = if (n == 10^valuation(n, 10), return (0)); k=1; while(#vecsort(digits(n^k),,8)!=10, k++); k; \\ Michel Marcus, Aug 20 2014

Python

def a(n):
  s = str(n)
  if n == 1 or (s.count('0')==len(s)-1 and s.startswith('1')):
    return 0
  k = 1
  count = 0
  while count != 10:
    count = 0
    for i in range(10):
      if str(n**k).count(str(i)) == 0:
        count += 1
        break
    if count:
      k += 1
    else:
      return k
n = 1
while n < 100:
  print(a(n), end=', ')
  n += 1
# Derek Orr, Aug 20 2014

Formula

a(10^e) = 0; a(m^e) = a(m)/e for e dividing a(m). - Reinhard Zumkeller, Dec 06 2004

Extensions

More terms from Reinhard Zumkeller, Dec 06 2004

Corrected a(15), a(17), a(38), a(48), a(56) and a(65). (For each of these terms, the only 1 in n^k is the first digit.) - Jon E. Schoenfield, Sep 20 2008