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A380885 a(n) is the smallest multiple m*n (m > 1) of n which contains every decimal digit of n, including repetitions.

%I A380885 #24 Feb 23 2025 11:19:08
%S A380885 10,12,30,24,15,36,70,48,90,100,110,120,130,140,105,160,170,108,190,
%T A380885 120,126,220,230,240,125,260,270,280,290,300,310,320,330,340,315,360,
%U A380885 370,380,390,240,164,294,344,440,405,460,470,384,294,150,153,520,530,540
%N A380885 a(n) is the smallest multiple m*n (m > 1) of n which contains every decimal digit of n, including repetitions.
%C A380885 Multiple m is in the range 3 <= m <= 10. Sequence is not the same as A087217, where digit order ("string") is important, whereas in this case it is not (however there are many common terms). The first composite departure is a(15) and the first prime departure is a(41); see Example.
%H A380885 Michael De Vlieger, <a href="/A380885/b380885.txt">Table of n, a(n) for n = 1..10000</a>
%H A380885 Michael De Vlieger, <a href="/A380885/a380885.png">Log log scatterplot of a(n)</a>, n = 1..10^5.
%F A380885 a(n) <= 10*n.
%e A380885 a(1) = 10 since 10 is the smallest multiple of 1 which contains every digit of 1.
%e A380885 a(15) = 7*15 = 105 since every digit of 15 is present in 105 (note A087217(15) = 150).
%e A380885 a(35) = 315 = 9*35 = A087217(35) because here digits 3 and 5 are in order.
%e A380885 a(41) = 4*41 = 164, the smallest multiple of 41 containing digits 1 and 4. This is the first prime departure from A087217, since A087217(41) = 410.
%t A380885 Reap[Do[d = DigitCount[n]; k = 2; While[! AllTrue[DigitCount[#] - d, # >= 0 &] &[n*k], k++]; Sow[k *= n], {n, 120}] ][[-1, 1]] (* _Michael De Vlieger_, Feb 20 2025 *)
%o A380885 (PARI) f(d) = vector(10, i, #select(x->(x==(i-1)), d));
%o A380885 isok(k, v) = my(w=f(digits(k))); for (i=1, 10, if (v[i] > w[i], return(0));); return(1);
%o A380885 a(n) = my(k=2*n, v=f(digits(n))); while(!isok(k, v), k+=n); k; \\ _Michel Marcus_, Feb 20 2025
%o A380885 (Python)
%o A380885 from collections import Counter
%o A380885 def a(n):
%o A380885     c = Counter(str(n))
%o A380885     return next(mn for mn in range(2*n, 11*n, n) if Counter(str(mn)) >= c)
%o A380885 print([a(n) for n in range(1, 55)]) # _Michael S. Branicky_, Feb 23 2025
%Y A380885 Cf. A087217.
%K A380885 nonn,base
%O A380885 1,1
%A A380885 _David James Sycamore_, Feb 07 2025
%E A380885 More terms from _Michel Marcus_, Feb 20 2025