A029790 None of the digits in k is present in k^2 or k^3.

2, 3, 7, 8, 22, 47, 53, 77, 92, 157, 187, 188, 192, 552, 558, 577, 707, 772, 922, 2522, 8338, 17177, 66888, 575757, 929522, 1717177, 8888588

Offset: 1

Author

Patrick De Geest

Keywords

• nonn
• base
• more
• hard

Comments

If it exists, a(28) > 10^9. - Derek Orr, Jul 13 2014

a(28) > 2 * 10^9. - Pratik Koirala, Eugene Fiorini, Otis Tweneboah, Nathan Fox, Jul 13 2015

From Manfred Scheucher, Jul 23 2015: (Start)

I strongly conjecture that a(28) does not exist.

If a(28) exists, then a(28) > 10^20.

(End) [Edited by Peter Munn, Feb 08 2024 and Manfred Scheucher, Feb 09 2024]

If it exists, a(28) > 10^37. - Michael S. Branicky, May 05 2024

Examples

			For k = 47, k^2 = 2209 and k^3 = 103823. 4 and 7 do not appear in either of these numbers.
		

Links

• Manfred Scheucher, corrected python script

Programs

Maple

filter:= proc(n) local S1,S23;
S1:= convert(convert(n,base,10),set);
S23:= convert(convert(n^2,base,10),set) union
      convert(convert(n^3,base,10),set);
nops(S1 intersect S23)=0
end proc:
select(filter,[$1..10^5]); # Robert Israel, Jul 14 2014

Mathematica

Select[Range@ 1000000, Intersection[IntegerDigits[#^2], IntegerDigits@ #] == {} && Intersection[IntegerDigits[#^3], IntegerDigits@ #] == {} &] (* Michael De Vlieger, Jul 23 2015 *)

PARI

isok(n) = d = vecsort(Set(digits(n))); dd = vecsort(Set(digits(n^2))); ddd = vecsort(Set(digits(n^3))); for (i=1, #d, if (vecsearch(dd, d[i]) || vecsearch(ddd, d[i]), return (0));); 1 \\ Michel Marcus, Jul 13 2014

PARI

is(n)=#setintersect(setunion(Set(digits(n^2)),Set(digits(n^3))), Set(digits(n)))==0 \\ Charles R Greathouse IV, Jul 23 2015

Python

def a(n):
  s = str(n)
  s2 = str(n**2)
  s3 = str(n**3)
  count = 0
  for i in s:
    if s2.count(i) == 0 and s3.count(i) == 0:
      count += 1
    else:
      break
  if count == len(s):
    return True
n = 1
while n < 10**9:
  if a(n):
    print(n, end=', ')
  n += 1
# Derek Orr, Jul 13 2014