A256786 Numbers which are divisible by prime(d) for all digits d in their decimal representation.

12, 14, 42, 55, 154, 222, 228, 714, 1122, 1196, 1212, 1414, 2112, 2142, 2262, 3355, 4144, 4242, 5335, 5544, 5555, 6162, 9499, 11112, 11144, 11214, 11424, 11466, 11622, 11818, 11914, 12222, 12882, 14112, 15554, 16666, 21216, 21222, 21252, 21888, 22122, 22212

Offset

1,1

Comments

All terms are zerofree, cf. A052382;

there is no term containing digits 1 and 3 simultaneously;

a(n) contains at least one digit 1 iff a(n) is even, cf. A011531, A005843;

a(n) contains at least one digit 2 iff a(n) mod 3 = 0, cf. A011532, A008585;

a(n) contains at least one digit 3 iff a(n) mod 10 = 5, cf. A011533, A017329;

A020639(a(n)) <= 23.

The equivalent in base 2 is the empty sequence, in base 3 it is A191681\{0}; see A256874-A256879 for the base 4 - base 9 variant, and A256870 for a variant where digits 0 are allowed but divisibility by prime(d+1) is required instead. - M. F. Hasler, Apr 11 2015

Links

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

Éric Angelini, Divisibility by primes, SeqFan list, Apr 10 2015.

Index entries for 10-automatic sequences.

Example

Smallest terms containing the nonzero decimal digits:
.  d | prime(d) |  n | a(n)
. ---+----------+--------------------------
.  1 |       2  |  1 |   12 = 2^2 * 3
.  2 |       3  |  1 |   12 = 2^2 * 3
.  3 |       5  | 16 | 3355 = 5 * 11 * 61
.  4 |       7  |  2 |   14 = 2 * 7
.  5 |      11  |  4 |   55 = 5 * 11
.  6 |      13  | 10 | 1196 = 2^2 * 13 * 23
.  7 |      17  |  8 |  714 = 2 * 3 * 7 * 17
.  8 |      19  |  7 |  228 = 2^2 * 3 * 19
.  9 |      23  | 10 | 1196 = 2^2 * 13 * 23 .

Mathematica

Select[Range@22222, FreeQ[IntegerDigits[#], 0]&&Total[Mod[#, Prime[IntegerDigits[#]]]]==0&] (* Ivan N. Ianakiev, Apr 11 2015 *)

Prog

(Haskell)
a256786 n = a256786_list !! (n-1)
a256786_list = filter f a052382_list where
   f x = g x where
     g z = z == 0 || x `mod` a000040 d == 0 && g z'
           where (z', d) = divMod z 10
(PARI) is_A256786(n)=!for(i=1, #d=Set(digits(n)), (!d[i]||n%prime(d[i]))&&return) \\ M. F. Hasler, Apr 11 2015
(Python)
primes = [1, 2, 3, 5, 7, 11, 13, 17, 19, 23]
def ok(n):
    s = str(n)
    return "0" not in s and all(n%primes[int(d)] == 0 for d in s)
print([k for k in range(22213) if ok(k)]) # Michael S. Branicky, Dec 14 2021

Crossrefs

Cf. A000040, A005843, A008585, A011531, A011532, A011533, A017329, A020639, A052382, A256874-A256879, A256882-A256884, A256865-A256870.

Sequence in context: A330197 A127401 A332876 * A337874 A337876 A058073

Adjacent sequences: A256783 A256784 A256785 * A256787 A256788 A256789

Keyword

nonn,base

Author

Eric Angelini and Reinhard Zumkeller, Apr 10 2015

Status

approved