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

1,2

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.

Links

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

Rémy Sigrist, PARI program

Example

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

Mathematica

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

Prog

(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

Crossrefs

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

Sequence in context: A246088 A071204 A002796 * A055471 A278328 A379262

Adjacent sequences: A383589 A383590 A383591 * A383593 A383594 A383595

Keyword

nonn,base,easy

Author

Rémy Sigrist, May 01 2025

Status

approved