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A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.

%I A330982 #45 Dec 23 2024 14:53:46
%S A330982 1,2,3,4,5,6,7,8,9,11,12,14,15,16,18,22,24,26,28,32,33,35,36,38,42,44,
%T A330982 46,48,52,54,55,62,64,66,68,72,74,76,77,82,84,86,88,92,94,96,98,99,
%U A330982 111,112,113,114,115,116,118,119,122,124,126,128,131,132,134,135,137,138,142,144,146,148,152
%N A330982 Remeven numbers: having an even remainder modulo any of their digits, digit 0 forbidden.
%C A330982 The sequence is a subset of the zeroless numbers A052382 which have asymptotic density 0 because they are in the complement of pandigital numbers A171102 which have asymptotic density 1. But does it have finite density within A052382?
%C A330982 It contains all repdigit numbers A010785 \ {0} and also all numbers with only even digits A014263 \ {0} and all numbers divisible by all of their digits, A034838.
%C A330982 The graph is self-similar, it looks the same whether we take the graph of values < 10^4 or that of values < 10^5 etc.: In the range 0 < a(n) < 10^(k+1), there are jumps of size > 10^k/9 where the values cross the limits d*10^k, 1 <= d <= 9 (from a(n) <= {d-1}9...9 to a(n+1) >= d1...1, since 0's are forbidden).
%C A330982 There are N = (0, 9, 48, 303, 2190, 15871, 119442, 930324, ...) terms below 10^k, k >= 0; these N(k) are also the indices of terms a(N(k)) = 10^k-1 (k>0), which are followed by repunits a(N(k)+1) = a(N(k+1))/9 (k >= 0).
%C A330982 The smallest zeroless pandigital term is a(8455060) = 123567894. - _Giovanni Resta_, Jan 08 2020
%H A330982 M. F. Hasler, <a href="/A330982/b330982.txt">Table of n, a(n) for n = 1..10000</a>, Jan 07 2020.
%H A330982 Eric Angelini, <a href="https://web.archive.org/web/*/http://list.seqfan.eu/oldermail/seqfan/2020-January/">Remeven numbers</a>, SeqFan list, Jan 05 2020.
%e A330982 12 is in the sequence because 12 % 1 = 0 and 12 % 2 = 0 both are even, where x % y is the remainder of x divided by y.
%e A330982 13 is not in the sequence because 13 % 3 = 1 is odd.
%t A330982 Select[Range[200],DigitCount[#,10,0]==0&&AllTrue[Mod[#,IntegerDigits[ #]],EvenQ]&] (* Requires Mathematica version 10 or later *) (* _Harvey P. Dale_, Apr 02 2020 *)
%o A330982 (PARI) select( {is_A330982(n,d=digits(n))=vecmin(d)&&!for(j=1,#d, bittest(n%d[j],0)&&return)}, [1..200]) \\ Using Set(digits) is about 20% slower
%o A330982 (Magma) [k:k in [1..160]|not 0 in Intseq(k) and forall{d:d in Intseq(k)|IsEven(k mod d)}]; // _Marius A. Burtea_, Jan 08 2020
%Y A330982 Cf. A330981 (remodd numbers).
%Y A330982 Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).
%K A330982 nonn,base
%O A330982 1,2
%A A330982 _Eric Angelini_ and _M. F. Hasler_, Jan 05 2020