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

Author

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

Keywords

• nonn
• base

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

Examples

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

Links

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

Crossrefs

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

Programs

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

Mathematica

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

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

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