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

1,1

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.

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.

Formula

a(n) <= 10*n.

Example

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.

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 *)

Prog

(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

Crossrefs

Cf. A087217.

Sequence in context: A129508 A015728 A080470 * A087217 A044972 A132313

Adjacent sequences: A380882 A380883 A380884 * A380886 A380887 A380888

Keyword

nonn,base

Author

David James Sycamore, Feb 07 2025

Extensions

More terms from Michel Marcus, Feb 20 2025

Status

approved