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.

12, 12, 36, 24, 15, 36, 147, 48, 189, 120, 132, 192, 312, 714, 105, 416, 714, 108, 912, 120, 126, 132, 322, 432, 125, 624, 702, 728, 928, 360, 713, 832, 132, 1394, 315, 936, 703, 836, 936, 240, 164, 294, 344, 264, 405, 644, 1457, 384, 294, 150, 153, 1352, 1325, 1458, 165, 1456, 3705, 1508, 295

Offset: 1

Author

Ali Sada and M. F. Hasler, Mar 04 2025

Keywords

• nonn
• base

Comments

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.

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

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

Examples

			36 is the least nontrivial multiple of 6 that contains the digit 6, so a(6) = 36.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 1..3500

Crossrefs

Cf. A380885 (without different start constraint).

Programs

Magma

A381700 := function(n)
    S := Set(Intseq(n));
    k := 1;
    repeat
        k +:= 1;
        digitsKn := Set(Intseq(k*n));
    until (S subset digitsKn) and (Max(Intseq(k)) ge 2);
    return k*n;
end function;
[ A381700(n) : n in [1..60] ]; // Vincenzo Librandi, Mar 17 2025

Mathematica

A381700[n_Integer]:=Module[{S,k,digitsKn},S=DeleteDuplicates[IntegerDigits[n]]; (* Set of digits of n *) k=1;
While[k++;
digitsKn=DeleteDuplicates[IntegerDigits[k n]];
!(SubsetQ[digitsKn,S]&&Max[IntegerDigits[k]]>=2)];
k n] (* Vincenzo Librandi, Mar 17 2025 *)

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));