A038770 Numbers divisible by at least one of their digits.

1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18, 19, 20, 21, 22, 24, 25, 26, 28, 30, 31, 32, 33, 35, 36, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 55, 60, 61, 62, 63, 64, 65, 66, 70, 71, 72, 75, 77, 80, 81, 82, 84, 85, 88, 90, 91, 92, 93, 95, 96, 99, 100, 101, 102

Offset

1,2

Comments

A038769(a(n)) > 0; complement of A038772.

The decimal digit strings of this sequence are a regular language, since it is the union of A011531 and A121022 .. A121029 which are likewise regular languages. Some computer state machine manipulation for this union shows a minimum deterministic finite automaton (DFA) matching the digit strings of this sequence has 11561 states. Reversing strings (so least significant digit first) reduces to 1699 states, or reverse and allow high 0's (which become trailing 0's due to the reverse) reduces to 1424 states. The latter are tractable sizes for the linear recurrence in A327560. - Kevin Ryde, Dec 04 2019

Links

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Index entries for 10-automatic sequences

Formula

a(n) ~ n. - Charles R Greathouse IV, Jul 22 2011

Example

35 is included because 5 is a divisor of 35, but 37 is not included because neither 3 nor 7 is a divisor of 37.

Mathematica

Select[Range[120], MemberQ[Divisible[#, Cases[IntegerDigits[#], Except[0]]], True]&] (* Harvey P. Dale, Jun 20 2011 *)
Select[Range[120], AnyTrue[#/DeleteCases[IntegerDigits[#], 0], IntegerQ]&] (* Harvey P. Dale, Mar 29 2024 *)

Prog

(Haskell)
a038770 n = a038770_list !! (n-1)
a038770_list = filter f [1..] where
   f u = g u where
     g v = v > 0 && (((d == 0 || r > 0) && g v') || r == 0)
           where (v', d) = divMod v 10; r = mod u d
-- Reinhard Zumkeller, Jul 30 2015, Jun 19 2011
(PARI) is(n)=my(v=vecsort(eval(Vec(Str(n))), , 8)); for(i=if(v[1], 1, 2), #v, if(n%v[i]==0, return(1))); 0 \\ Charles R Greathouse IV, Jul 22 2011
(Python)
def ok(n): return any(n%int(d) == 0 for d in str(n) if d != '0')
print(list(filter(ok, range(1, 103)))) # Michael S. Branicky, May 20 2021

Crossrefs

Cf. A327560 (counts), A038772 (complement), A034709, A034837, A038769.

Sequence in context: A359415 A080682 A182049 * A342591 A193176 A263314

Adjacent sequences: A038767 A038768 A038769 * A038771 A038772 A038773

Keyword

base,easy,nonn,nice

Author

Henry Bottomley, May 04 2000

Status

approved