cp's OEIS Frontend

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

%I A266586 #26 Nov 15 2022 12:20:39
%S A266586 1,6163,51,416,251,21,967,86,255,1,1,1,1,1,1,1,1,1,1,1255,1,781,973,
%T A266586 26,265,24,81,1139,1135,51,1,291,186,151,41,936,3001,886,982,416,1,
%U A266586 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
%N A266586 The least nonnegative integer N such that n*N has the same digits as n and N together, not counting repetitions.
%C A266586 See A266578 for the variant where repeated digits are counted.
%C A266586 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, ...).
%C A266586 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, ...
%C A266586 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.
%H A266586 M. F. Hasler, <a href="/A266586/b266586.txt">Table of n, a(n) for n = 1..10000</a>
%F A266586 a(n) = 1 whenever n has a digit '1', i.e., n in A011531.
%F A266586 a(n) <= A266578(n) unless A266578(n) = 0.
%e A266586 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.
%p A266586 f:= proc(n) local k,Ln,Lk,Lnk;
%p A266586   Ln:= convert(convert(n,base,10),set);
%p A266586   if has(Ln,1) then return 1 fi;
%p A266586   for k from 2 do
%p A266586     Lk:= convert(convert(k,base,10),set);
%p A266586     Lnk:= convert(convert(n*k,base,10),set);
%p A266586     if Lnk = Ln union Lk then return k fi
%p A266586   od
%p A266586 end proc:
%p A266586 map(f, [$1..100]); # _Robert Israel_, Jan 01 2016
%o A266586 (PARI) A266586(n,L=9e9,d=digits(n))=for(k=1,L,Set(digits(k*n))==Set(concat(digits(k),d))&&return(k))
%o A266586 (Python)
%o A266586 from itertools import count
%o A266586 def a(n):
%o A266586     digs = set(str(n))
%o A266586     return next(N for N in count(1) if digs | set(str(N)) == set(str(n*N)))
%o A266586 print([a(n) for n in range(1, 67)]) # _Michael S. Branicky_, Nov 15 2022
%Y A266586 Cf. A266578, A266798.
%K A266586 nonn,base
%O A266586 1,2
%A A266586 _M. F. Hasler_, Jan 01 2016