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

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, 46, 48, 52, 54, 55, 62, 64, 66, 68, 72, 74, 76, 77, 82, 84, 86, 88, 92, 94, 96, 98, 99, 111, 112, 113, 114, 115, 116, 118, 119, 122, 124, 126, 128, 131, 132, 134, 135, 137, 138, 142, 144, 146, 148, 152

Offset: 1

Author

Eric Angelini and M. F. Hasler, Jan 05 2020

Keywords

• nonn
• base

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

Examples

			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.
		

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.

Crossrefs

Cf. A330981 (remodd numbers).

Cf. A171102 (pandigitals), A010785 (repdigits), A014263 (only even digits), A034838 (divisible by all digits).

Programs

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

Mathematica

Select[Range[200],DigitCount[#,10,0]==0&&AllTrue[Mod[#,IntegerDigits[ #]],EvenQ]&] (* Requires Mathematica version 10 or later *) (* Harvey P. Dale, Apr 02 2020 *)

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