

A298980


Numbers n such that there exists an integer k < n for which the significant decimal digits of k/n (i.e., neglecting leading zeros) are those of n.


3



3, 6, 7, 8, 10, 14, 17, 20, 22, 26, 28, 30, 33, 36, 37, 40, 41, 42, 50, 57, 58, 59, 60, 62, 63, 64, 66, 67, 69, 70, 71, 72, 73, 74, 76, 77, 78, 79, 80, 81, 82, 83, 84, 85, 86, 87, 88, 89, 90, 91, 92, 93, 94, 95, 96, 97, 98, 100, 114, 118, 122, 126, 130, 134, 141, 148, 158, 161, 164, 167, 170, 173, 176, 184, 187
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OFFSET

1,1


COMMENTS

Otherwise said, floor(10^m*k/n) = n for some k and m.
Also, numbers n which have n as a subsequence in the decimal expansion of k/n, 0 < k < n.
Initially it appears that if n is present so is 10n and 11n. These two statements are false. 14 is present but 140 is not. 1/140 = 0.00714285... 17 is present but 187 is not.
However if there is a k between 0 and n so that the GCD(k,n) = r > 1 and k/r is used to show that n/r is a member, then so is n. As an example, 33 is a member since 11/33 = 1/3 and 3 is a member. See the first example.
The density of numbers in this sequence appears to increase to above 55% near n ~ 10^9. See A298981 for the complement and A298982 for the kvalues.


LINKS

Table of n, a(n) for n=1..75.


EXAMPLE

3 is a member since 1/3 = 0.3333... and this decimal number begins with 3;
6 is a member since 10/6 = 1.666... and its decimal part begins with 6;
7 is a member since 5/7 = 0.714285... and its decimal part begins with 7;
8 is a member since 7/8 = 0.87500... and its decimal part begins with 8;
10 is a member since 1/10 = 0.1000... and its decimal part begins with 10;
14 is a member since 2/14 = 0.142857... and its decimal part begins with 14;
17 is a member since 3/17 = 0.17647058823 ... and its decimal part begins with 17; etc.


MATHEMATICA

fQ = Compile[{{n, _Integer}}, Block[{k = 1, il = IntegerLength@ n}, While[m = 10^il*k/n; While[ IntegerLength@ Floor@ m < il, m *= 10]; k < n && Floor[m] != n, k++]; k < n]]; Select[Range@200, fQ]


PROG

(PARI) is_A298980(n, k=(n^21)\10^(logint(n, 10)+1)+1)={k*10^(logint((n^2(n>1))\k, 10)+1)\n==n} \\ Or use A298982 to get the kvalue if n is in this sequence or 0 else. \\ M. F. Hasler, Feb 01 2018


CROSSREFS

Inspired by and equal to the range (= sorted terms) of A298232.
Complement of A298981.
Cf. A051626, A298982.
Sequence in context: A266986 A047283 A155932 * A206586 A289176 A277851
Adjacent sequences: A298977 A298978 A298979 * A298981 A298982 A298983


KEYWORD

easy,nonn


AUTHOR

Eric Angelini, M. F. Hasler and Robert G. Wilson v, Jan 30 2018


STATUS

approved



