A256786 - OEIS

%I #35 Jan 02 2023 12:30:51

%S 12,14,42,55,154,222,228,714,1122,1196,1212,1414,2112,2142,2262,3355,

%T 4144,4242,5335,5544,5555,6162,9499,11112,11144,11214,11424,11466,

%U 11622,11818,11914,12222,12882,14112,15554,16666,21216,21222,21252,21888,22122,22212

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

%C All terms are zerofree, cf. A052382;

%C there is no term containing digits 1 and 3 simultaneously;

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

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

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

%C A020639(a(n)) <= 23.

%C 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

%H Lars Blomberg and Reinhard Zumkeller, <a href="/A256786/b256786.txt">Table of n, a(n) for n = 1..10000</a>

%H Éric Angelini, <a href="http://list.seqfan.eu/oldermail/seqfan/2015-April/014678.html">Divisibility by primes</a>, SeqFan list, Apr 10 2015.

%H <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.

%e Smallest terms containing the nonzero decimal digits:

%e .  d | prime(d) |  n | a(n)

%e . ---+----------+--------------------------

%e .  1 |       2  |  1 |   12 = 2^2 * 3

%e .  2 |       3  |  1 |   12 = 2^2 * 3

%e .  3 |       5  | 16 | 3355 = 5 * 11 * 61

%e .  4 |       7  |  2 |   14 = 2 * 7

%e .  5 |      11  |  4 |   55 = 5 * 11

%e .  6 |      13  | 10 | 1196 = 2^2 * 13 * 23

%e .  7 |      17  |  8 |  714 = 2 * 3 * 7 * 17

%e .  8 |      19  |  7 |  228 = 2^2 * 3 * 19

%e .  9 |      23  | 10 | 1196 = 2^2 * 13 * 23 .

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

%o (Haskell)

%o a256786 n = a256786_list !! (n-1)

%o a256786_list = filter f a052382_list where

%o    f x = g x where

%o      g z = z == 0 || x `mod` a000040 d == 0 && g z'

%o            where (z', d) = divMod z 10

%o (PARI) is_A256786(n)=!for(i=1,#d=Set(digits(n)),(!d[i]||n%prime(d[i]))&&return) \\ _M. F. Hasler_, Apr 11 2015

%o (Python)

%o primes = [1, 2, 3, 5, 7, 11, 13, 17, 19, 23]

%o def ok(n):

%o     s = str(n)

%o     return "0" not in s and all(n%primes[int(d)] == 0 for d in s)

%o print([k for k in range(22213) if ok(k)]) # _Michael S. Branicky_, Dec 14 2021

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

%K nonn,base

%O 1,1

%A _Eric Angelini_ and _Reinhard Zumkeller_, Apr 10 2015