OFFSET
1,2
COMMENTS
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?
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.
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).
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).
The smallest zeroless pandigital term is a(8455060) = 123567894. - Giovanni Resta, Jan 08 2020
LINKS
M. F. Hasler, Table of n, a(n) for n = 1..10000, Jan 07 2020.
Eric Angelini, Remeven numbers, SeqFan list, Jan 05 2020.
EXAMPLE
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.
13 is not in the sequence because 13 % 3 = 1 is odd.
MATHEMATICA
Select[Range[200], DigitCount[#, 10, 0]==0&&AllTrue[Mod[#, IntegerDigits[ #]], EvenQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 02 2020 *)
PROG
(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
(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
CROSSREFS
KEYWORD
nonn,base
AUTHOR
Eric Angelini and M. F. Hasler, Jan 05 2020
STATUS
approved