A383592 Positive integers k divisible by all positive integers whose decimal expansion appears as a substring of k.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 15, 20, 22, 24, 30, 33, 36, 40, 44, 48, 50, 55, 60, 66, 70, 77, 80, 88, 90, 99, 100, 110, 120, 150, 200, 210, 220, 240, 250, 300, 330, 360, 400, 420, 440, 480, 500, 510, 520, 550, 600, 630, 660, 700, 770, 800, 840, 880

Offset: 1

Author

Rémy Sigrist, May 01 2025

Keywords

• nonn
• base
• easy

Comments

This sequence is infinite as ten times a term is also a term.

All terms are of the form A037124(k) or A037124(k) + d where k > 0 and d divides A037124(k) while having strictly less decimal digits as A037124(k).

Empirically, all terms have either one or two nonzero decimal digits.

Examples

			The number 240 is divisible by 2, 24, 240, 4 and 40, so 240 belongs to this sequence.
		

Links

• Rémy Sigrist, Table of n, a(n) for n = 1..10314
• Rémy Sigrist, PARI program

Crossrefs

Cf. A037124, A078546, A175381 (binary variant), A178157, A218978.

Programs

Mathematica

Select[Range[880],AllTrue[#/Select[FromDigits/@Subsequences[IntegerDigits[#]],#>0&],IntegerQ]&] (* James C. McMahon, May 13 2025 *)

PARI

is(n, base = 10) = {
    my (d = digits(n, base));
    for (i = 1, #d,
        if (d[i],
            for (j = i, #d,
                if (n % fromdigits(d[i..j], base),
                    return (0);););););
    return (1); }

PARI

\\ See Links section.

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 all(n%v == 0 for ss in subs if (v:=int(ss)) > 0)
print([k for k in range(1000) if ok(k)]) # Michael S. Branicky, May 09 2025