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

1,2

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

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)

Formula

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

Example

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

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 *)

Prog

(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

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).

Sequence in context: A196109 A196106 A171744 * A259081 A143737 A214601

Adjacent sequences: A090490 A090491 A090492 * A090494 A090495 A090496

Keyword

base,nonn

Author

Amarnath Murthy, Dec 03 2003

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

Status

approved