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

%I A256786 #36 Dec 23 2024 14:53:44
%S A256786 12,14,42,55,154,222,228,714,1122,1196,1212,1414,2112,2142,2262,3355,
%T A256786 4144,4242,5335,5544,5555,6162,9499,11112,11144,11214,11424,11466,
%U A256786 11622,11818,11914,12222,12882,14112,15554,16666,21216,21222,21252,21888,22122,22212
%N A256786 Numbers which are divisible by prime(d) for all digits d in their decimal representation.
%C A256786 All terms are zerofree, cf. A052382;
%C A256786 there is no term containing digits 1 and 3 simultaneously;
%C A256786 a(n) contains at least one digit 1 iff a(n) is even, cf. A011531, A005843;
%C A256786 a(n) contains at least one digit 2 iff a(n) mod 3 = 0, cf. A011532, A008585;
%C A256786 a(n) contains at least one digit 3 iff a(n) mod 10 = 5, cf. A011533, A017329;
%C A256786 A020639(a(n)) <= 23.
%C A256786 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 A256786 Lars Blomberg and Reinhard Zumkeller, <a href="/A256786/b256786.txt">Table of n, a(n) for n = 1..10000</a>
%H A256786 Éric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2015-April/014678.html">Divisibility by primes</a>, SeqFan list, Apr 10 2015.
%H A256786 <a href="/index/Ar#10-automatic">Index entries for 10-automatic sequences</a>.
%e A256786 Smallest terms containing the nonzero decimal digits:
%e A256786 .  d | prime(d) |  n | a(n)
%e A256786 . ---+----------+--------------------------
%e A256786 .  1 |       2  |  1 |   12 = 2^2 * 3
%e A256786 .  2 |       3  |  1 |   12 = 2^2 * 3
%e A256786 .  3 |       5  | 16 | 3355 = 5 * 11 * 61
%e A256786 .  4 |       7  |  2 |   14 = 2 * 7
%e A256786 .  5 |      11  |  4 |   55 = 5 * 11
%e A256786 .  6 |      13  | 10 | 1196 = 2^2 * 13 * 23
%e A256786 .  7 |      17  |  8 |  714 = 2 * 3 * 7 * 17
%e A256786 .  8 |      19  |  7 |  228 = 2^2 * 3 * 19
%e A256786 .  9 |      23  | 10 | 1196 = 2^2 * 13 * 23 .
%t A256786 Select[Range@22222,FreeQ[IntegerDigits[#],0]&&Total[Mod[#,Prime[IntegerDigits[#]]]]==0&] (* _Ivan N. Ianakiev_, Apr 11 2015 *)
%o A256786 (Haskell)
%o A256786 a256786 n = a256786_list !! (n-1)
%o A256786 a256786_list = filter f a052382_list where
%o A256786    f x = g x where
%o A256786      g z = z == 0 || x `mod` a000040 d == 0 && g z'
%o A256786            where (z', d) = divMod z 10
%o A256786 (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 A256786 (Python)
%o A256786 primes = [1, 2, 3, 5, 7, 11, 13, 17, 19, 23]
%o A256786 def ok(n):
%o A256786     s = str(n)
%o A256786     return "0" not in s and all(n%primes[int(d)] == 0 for d in s)
%o A256786 print([k for k in range(22213) if ok(k)]) # _Michael S. Branicky_, Dec 14 2021
%Y A256786 Cf. A000040, A005843, A008585, A011531, A011532, A011533, A017329, A020639, A052382, A256874-A256879, A256882-A256884, A256865-A256870.
%K A256786 nonn,base
%O A256786 1,1
%A A256786 _Eric Angelini_ and _Reinhard Zumkeller_, Apr 10 2015