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A381700 a(n) is the smallest multiple k*n that contains all the distinct digits of n, where k > 0 must have at least one digit > 1.

%I A381700 #47 Apr 29 2025 13:20:21
%S A381700 12,12,36,24,15,36,147,48,189,120,132,192,312,714,105,416,714,108,912,
%T A381700 120,126,132,322,432,125,624,702,728,928,360,713,832,132,1394,315,936,
%U A381700 703,836,936,240,164,294,344,264,405,644,1457,384,294,150,153,1352,1325,1458,165,1456,3705,1508,295
%N A381700 a(n) is the smallest multiple k*n that contains all the distinct digits of n, where k > 0 must have at least one digit > 1.
%C A381700 The restriction on k is to avoid trivial solutions where k is a sum of distinct powers of 10. Even if we exclude that k*n starts with the digits of n, most terms would correspond to k = 10^L+1 where L is about half the number of digits of n, so that the initial and final digits of the product are "trivially" those of n.
%C A381700 Asymptotically 100% of positive integers n have a(n) = 2*n. In particular, for n <= x, a(n) > 2*n occurs at most O(x^k) where k = log 9/log 10 = 0.954.... - _Charles R Greathouse IV_, Mar 20 2025
%C A381700 Is a(n)/n bounded? Define b(n) = a(n)/n; then b(42941736805) = 5609 is the largest I know. - _Charles R Greathouse IV_, Apr 24 2025
%H A381700 Vincenzo Librandi, <a href="/A381700/b381700.txt">Table of n, a(n) for n = 1..3500</a>
%e A381700 36 is the least nontrivial multiple of 6 that contains the digit 6, so a(6) = 36.
%t A381700 A381700[n_Integer]:=Module[{S,k,digitsKn},S=DeleteDuplicates[IntegerDigits[n]]; (* Set of digits of n *) k=1;
%t A381700 While[k++;
%t A381700 digitsKn=DeleteDuplicates[IntegerDigits[k n]];
%t A381700 !(SubsetQ[digitsKn,S]&&Max[IntegerDigits[k]]>=2)];
%t A381700 k n] (* _Vincenzo Librandi_, Mar 17 2025 *)
%o A381700 (PARI) A381700(n)=my(S=Set(digits(n))); for(k=2,oo, #setminus(S,Set(digits(k*n))) || vecmax(digits(k))<2 || return(k*n));
%o A381700 (Magma)
%o A381700 A381700 := function(n)
%o A381700     S := Set(Intseq(n));
%o A381700     k := 1;
%o A381700     repeat
%o A381700         k +:= 1;
%o A381700         digitsKn := Set(Intseq(k*n));
%o A381700     until (S subset digitsKn) and (Max(Intseq(k)) ge 2);
%o A381700     return k*n;
%o A381700 end function;
%o A381700 [ A381700(n) : n in [1..60] ]; // _Vincenzo Librandi_, Mar 17 2025
%Y A381700 Cf. A380885 (without different start constraint).
%K A381700 nonn,base
%O A381700 1,1
%A A381700 _Ali Sada_ and _M. F. Hasler_, Mar 04 2025