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

10, 12, 30, 24, 15, 36, 70, 48, 90, 100, 110, 120, 130, 140, 105, 160, 170, 108, 190, 120, 126, 220, 230, 240, 125, 260, 270, 280, 290, 300, 310, 320, 330, 340, 315, 360, 370, 380, 390, 240, 164, 294, 344, 440, 405, 460, 470, 384, 294, 150, 153, 520, 530, 540

Offset: 1

Author

David James Sycamore, Feb 07 2025

Keywords

• nonn
• base

Comments

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.

Examples

			a(1) = 10 since 10 is the smallest multiple of 1 which contains every digit of 1.
a(15) = 7*15 = 105 since every digit of 15 is present in 105 (note A087217(15) = 150).
a(35) = 315 = 9*35 = A087217(35) because here digits 3 and 5 are in order.
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.
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..10000
• Michael De Vlieger, Log log scatterplot of a(n), n = 1..10^5.

Crossrefs

Cf. A087217.

Programs

Mathematica

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 *)

PARI

f(d) = vector(10, i, #select(x->(x==(i-1)), d));
isok(k, v) = my(w=f(digits(k))); for (i=1, 10, if (v[i] > w[i], return(0));); return(1);
a(n) = my(k=2*n, v=f(digits(n))); while(!isok(k, v), k+=n); k; \\ Michel Marcus, Feb 20 2025

Python

from collections import Counter
def a(n):
    c = Counter(str(n))
    return next(mn for mn in range(2*n, 11*n, n) if Counter(str(mn)) >= c)
print([a(n) for n in range(1, 55)]) # Michael S. Branicky, Feb 23 2025

Formula

a(n) <= 10*n.

Extensions

More terms from Michel Marcus, Feb 20 2025