A298232 -id:A298232 - cp's OEIS frontend

A298982 a(n) is the least k for which the most significant decimal digits of k/n (disregarding any leading zeros) are n, or 0 if no such k exists.

0, 0, 1, 0, 0, 4, 5, 7, 0, 1, 0, 0, 0, 2, 0, 0, 3, 0, 0, 4, 0, 5, 0, 0, 0, 7, 0, 8, 0, 9, 0, 0, 11, 0, 0, 13, 14, 0, 0, 16, 17, 18, 0, 0, 0, 0, 0, 0, 0, 25, 0, 0, 0, 0, 0, 0, 33, 34, 35, 36, 0, 39, 4, 41, 0, 44, 45, 0, 48, 49, 51, 52, 54, 55, 0, 58, 6, 61, 63, 64, 66, 68, 69, 71, 73, 74, 76, 78, 8, 81, 83, 85, 87, 89, 91, 93, 95, 97, 0, 1

Offset: 1

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

By decimal digits we mean those of the fractional part of k/n. Otherwise said, we require floor(10^m*k/n) = n for some k < n and m.
Indices of 0's are listed in A298981, indices of the other terms are listed in A298980.
It appears that the asymptotic density of 0's is slightly below 45%: The number of 0's among a(1..10^k) is (5, 42, 461, 4553, 45423, 451315, 4506142, 45017570, ...). Is there a simple estimate for the exact value? - M. F. Hasler, Feb 01 2018
There may be no asymptotic density: the fraction of 0's fluctuates too much.  See the linked plot.

			a(1) = 0 since there does not exist any k such that k/1 has a decimal digit which begins with 1 (cf. comment).
a(6) = 4 since 4/6 = 0.666... and its decimal digit begins with 6.
a(28) = 8 since 8/28 = 0.28571428571428... even though 1/28 = 0.0357142857142857... has "28" as a subsequence.

M. F. Hasler, Table of n, a(n) for n = 1..10000
Robert Israel, Plot: Fraction of 0's in a(1) to a(n)

Cf. A298232, A051626, A298980, A298981.

f:= proc (n) local m, k;
  for m from ceil(log[10](n^2)) by -1 to 1 do
     k := ceil(n^2/10^m);
     if n <= k then return 0 end if;
     if k < n*(n+1)/10^m then return k end if
  end do;
  0
end proc:
map(f, [$1..200]); # Robert Israel, Feb 09 2018

f = 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++]; If[k < n, k, 0]]]; Array[f, 100]

A298982(n,k=(n^2-1)\10^(logint(n,10)+1)+1)={k*10^(logint((n^2-(n>1))\k, 10)+1)\n==n && return(k\10^valuation(k,10))} \\ M. F. Hasler, Feb 01 2018

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.

Original entry on oeis.org

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

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

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.

			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.

Inspired by and equal to the range (= sorted terms) of A298232.
Complement of A298981.
Cf. A051626, A298982.

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]

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

A298981 Numbers m such that there does not exist an integer k < m for which the initial decimal digits of k/m are m.

Original entry on oeis.org

1, 2, 4, 5, 9, 11, 12, 13, 15, 16, 18, 19, 21, 23, 24, 25, 27, 29, 31, 32, 34, 35, 38, 39, 43, 44, 45, 46, 47, 48, 49, 51, 52, 53, 54, 55, 56, 61, 65, 68, 75, 99, 101, 102, 103, 104, 105, 106, 107, 108, 109, 110, 111, 112, 113, 115, 116, 117, 119, 120, 121, 123, 124, 125, 127, 128, 129, 131, 132, 133, 135, 136, 137, 138, 139

Offset: 1

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

Inspired by A298232.

			2 is in the sequence since there is no k such that k/2 would result in a decimal number which begins with 2, i.e., 0.2000. Instead, the decimal number for odd k's begin with 0.5.

Cf. A298232, A051626, A298980, A298982.

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@140, fQ]

PARI
```
is_A298981(n)=!A298982(n)
```

Complement of A298980.

cp's OEIS Frontend

A298982 a(n) is the least k for which the most significant decimal digits of k/n (disregarding any leading zeros) are n, or 0 if no such k exists.

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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.

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Mathematica

PARI

A298981 Numbers m such that there does not exist an integer k < m for which the initial decimal digits of k/m are m.

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Mathematica

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