A266586 The least nonnegative integer N such that n*N has the same digits as n and N together, not counting repetitions.

1, 6163, 51, 416, 251, 21, 967, 86, 255, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1255, 1, 781, 973, 26, 265, 24, 81, 1139, 1135, 51, 1, 291, 186, 151, 41, 936, 3001, 886, 982, 416, 1, 341, 315, 1464, 181, 734, 371, 958, 1921, 251, 1, 2412, 635, 846, 221, 1801, 125, 948, 845, 21, 1, 251, 585, 2213, 281, 1076

Offset: 1

Author

M. F. Hasler, Jan 01 2016

Keywords

• nonn
• base

Comments

See A266578 for the variant where repeated digits are counted.

a(n) = 1 for 100 <= n <= 199 (and whenever n has a digit 1, cf. A011531), but then the sequence continues nontrivially with a(200,...) = (1255, 1, 751, 621, 251, 99, 511, 97, 101, 101, ...).

Record values are a(2) = 6163, a(2953) = 6521, a(3597) = 7209, a(5904) = 8047, a(23222) = 7681, a(39808) = 8011, a(39993) = 8231, a(44444) = 10151, ...

For small k=1,...,6, the graphs over the range 1 .. 10^(k+1) are roughly ("self"-)similar, because of the ranges 10^k .. 2*10^k-1 and (m+1/10)*10^k .. (m+2/10)*10^k-1 (with m=2,...,9) etc., on which a(n) = 1, while generically a(n) has values ranging quite uniformly between 1 and 10^4. For larger k, the picture changes, since pandigital numbers (and therefore also numbers having a digit '1') have asymptotic density one.

Examples

			a(2) = 6163 since 2*6163 = 12326 has the same digits (1, 2, 3 and 6) as concat(2,6163) = 26163, and 6163 is the least N with this property.
		

Links

• M. F. Hasler, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A266578, A266798.

Programs

Maple

f:= proc(n) local k,Ln,Lk,Lnk;
  Ln:= convert(convert(n,base,10),set);
  if has(Ln,1) then return 1 fi;
  for k from 2 do
    Lk:= convert(convert(k,base,10),set);
    Lnk:= convert(convert(n*k,base,10),set);
    if Lnk = Ln union Lk then return k fi
  od
end proc:
map(f, [$1..100]); # Robert Israel, Jan 01 2016

PARI

A266586(n,L=9e9,d=digits(n))=for(k=1,L,Set(digits(k*n))==Set(concat(digits(k),d))&&return(k))

Python

from itertools import count
def a(n):
    digs = set(str(n))
    return next(N for N in count(1) if digs | set(str(N)) == set(str(n*N)))
print([a(n) for n in range(1, 67)]) # Michael S. Branicky, Nov 15 2022

Formula

a(n) = 1 whenever n has a digit '1', i.e., n in A011531.

a(n) <= A266578(n) unless A266578(n) = 0.