A330562 Positive numbers k with property that if d is any nonzero digit of k then k mod d is also a digit of k.

10, 20, 30, 40, 50, 60, 70, 80, 90, 100, 101, 102, 103, 104, 105, 109, 110, 120, 130, 140, 150, 190, 200, 201, 202, 204, 206, 208, 210, 220, 230, 240, 250, 260, 280, 290, 300, 301, 302, 303, 306, 309, 310, 320, 330, 360, 390, 400, 401, 402, 404, 408, 420, 440, 460, 480, 500, 501, 502, 504, 505, 510, 520, 540, 550, 590

Offset

1,1

Comments

Theorem: k must have a zero digit.

Proof: If not, let s be the smallest digit in k. Then d = (k mod s) is a digit of k, and d < s. Contradiction.

Pandigital numbers (A171102) are necessarily an infinite subset. - Hans Havermann, Jan 02 2020

Links

Rémy Sigrist, Table of n, a(n) for n = 1..25000

Example

401 is a term since 401 mod 4 = 1 and 401 mod 1 = 0, and 1 and 0 are both digits of 401.

Mathematica

Select[Range@ 600, Function[{k, d}, AllTrue[DeleteCases[d, 0], ! FreeQ[d, Mod[k, #]] &]] @@ {#, IntegerDigits[#]} &] (* Michael De Vlieger, Jan 01 2020 *)

Prog

(PARI) is(k) = my (d=Set(digits(k))); for (i=1, #d, if (d[i] && setsearch(d, k%d[i])==0, return (0))); return (1) \\ Rémy Sigrist, Jan 01 2020
(Magma) [k:k in [1..600]| forall{c:c in Set(Intseq(k)) diff {0}| k mod c in Intseq(k)}]; // Marius A. Burtea, Jan 01 2020
(Python)
def ok(n): s = set(map(int, str(n))); return all(n%d in s for d in s-{0})
print([k for k in range(1, 600) if ok(k)]) # Michael S. Branicky, Dec 23 2024

Crossrefs

Cf. A330563 (primes), A171102 (pandigital subset).

Sequence in context: A011540 A098394 A057169 * A328783 A342855 A201014

Adjacent sequences: A330559 A330560 A330561 * A330563 A330564 A330565

Keyword

nonn,base

Author

N. J. A. Sloane, Dec 31 2019, following a suggestion from Eric Angelini

Status

approved