A383749 Positive numbers k whose decimal expansion does not contain the decimal expansion of any proper divisor of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 23, 27, 29, 34, 37, 38, 43, 46, 47, 49, 53, 54, 56, 57, 58, 59, 67, 68, 69, 73, 74, 76, 78, 79, 83, 86, 87, 89, 94, 97, 98, 203, 207, 209, 223, 227, 229, 233, 239, 247, 249, 253, 257, 259, 263, 267, 269, 277, 283, 289, 293, 299, 307

Offset

1,2

Comments

Also fixed points of A121042.

This sequence is infinite as it contains A173041.

a(n) > 5 contains no decimal digit 1 and does not end in 2 or 5. - Michael S. Branicky, May 11 2025

Links

Rémy Sigrist, Table of n, a(n) for n = 1..10000

Example

The proper divisors of 54 are 1, 2, 3, 6, 9, 18 and 27; none of them appear in the decimal expansion of 54 so 54 belongs to this sequence.

Mathematica

A383749Q[k_] := SelectFirst[Divisors[k], StringContainsQ[IntegerString[k], IntegerString[#]] &] == k;
Select[Range[500], A383749Q] (* Paolo Xausa, May 12 2025 *)

Prog

(PARI) is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if ((i!=1 || j!=#d) && n % fromdigits(d[i..j], base)==0,
                    return (0); ); ); ); );
    return (1); }
(Python)
from itertools import count, islice
from sympy import divisors
def A383749_gen(startvalue=1): # generator of terms >= startvalue
    return filter(lambda n:not any(d<n and str(d) in str(n) for d in divisors(n, generator=True)), count(max(startvalue, 1)))
A383749_list = list(islice(A383749_gen(), 40)) # Chai Wah Wu, May 10 2025
(Python)
def ok(n):
    s = str(n)
    subs = (s[i:j] for i in range(len(s)) for j in range(i+1, len(s)+1) if s[i]!='0')
    return n and not any(n%v == 0 for ss in subs if n > (v:=int(ss)) > 0)
print([k for k in range(308) if ok(k)]) # Michael S. Branicky, May 11 2025

Crossrefs

Cf. A011531, A027751, A038772, A121042, A383592.

A038603 and A173041 are subsequences.

Sequence in context: A061672 A272814 A322516 * A132080 A307636 A373722

Adjacent sequences: A383746 A383747 A383748 * A383750 A383751 A383752

Keyword

nonn,base

Author

Rémy Sigrist, May 08 2025

Status

approved