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, 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

Offset: 1

Author

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

Keywords

• easy
• base
• nonn

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 gcd(k,n) = r > 1 and k/r is used to show that n/r is a term, then so is n. As an example, 33 is a term since 11/33 = 1/3 and 3 is a term. 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 k-values.

Examples

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

Crossrefs

Inspired by and equal to the range (= sorted terms) of A298232.

Complement of A298981.

Cf. A051626, A298982.

Programs

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]

PARI

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