cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-5 of 5 results.

A276466 a(n) = numerator of Sum_{d|n} 0.d.

Original entry on oeis.org

1, 3, 2, 7, 3, 6, 4, 3, 13, 9, 21, 43, 23, 57, 21, 83, 27, 57, 29, 3, 131, 63, 33, 69, 17, 69, 157, 91, 39, 9, 41, 99, 21, 81, 33, 79, 47, 87, 23, 27, 51, 267, 53, 147, 12, 99, 57, 17, 129, 33, 27, 161, 63, 309, 63, 159, 29, 117, 69, 357, 71, 123, 71, 131, 69
Offset: 1

Views

Author

Jaroslav Krizek, Sep 04 2016

Keywords

Comments

Let d be a divisor of n; 0.d means the decimal fraction formed by writing d after the decimal point, e.g., 0.12 = 12/100 = 3/25.
The first few values of Sum_{d|n} 0.d for n = 1,2,.. are 1/10, 3/10, 2/5, 7/10, 3/5, 6/5, 4/5, 3/2, 13/10, 9/10, 21/100, 43/25, ...
16450 is the only number < 5*10^7 such that Sum_{d|n} 0.d is an integer: Sum_{d | 16450} 0.d = 0.1 + 0.2 + 0.5 + 0.7 + 0.10 + 0.14 + 0.25 + 0.35 + 0.47 + 0.50 + 0.70 + 0.94 + 0.175 + 0.235 + 0.329 + 0.350 + 0.470 + 0.658 + 0.1175 + 0.1645 + 0.2350 + 0.3290 + 0.8225 + 0.16450 = 9; see A276465.
No other term like 16450 up to 4*10^11. - Giovanni Resta, Apr 03 2019

Examples

			For n=12; Sum_{d | 12} 0.d = 0.1 + 0.2 + 0.3 + 0.4 + 0.6 + 0.12 = 1.72 = 172/100 = 43/25; a(12) = 43.

Links

Crossrefs

Cf. A276465, A276467 (denominator).
Cf. A078267 and A078268 (both for 0.d).

Programs

  • Magma
    [Numerator(&+[d / (10^(#Intseq(d))): d in Divisors(n)]): n in [1..1000]]
  • Mathematica
    Table[Numerator@ Total@ (#*1/10^(1 + Floor@ Log10@ #)) &@ Divisors@ n, {n, 65}] (* Michael De Vlieger, Sep 04 2016 *)
  • PARI
    a(n) = numerator(sumdiv(n, d, d/10^(#Str(d)))); \\ Michel Marcus, Mar 29 2019
  • Python
    from fractions import Fraction
from sympy import divisors
def A276466(n):
    return sum(Fraction(d,10**len(str(d))) for d in divisors(n)).numerator # Chai Wah Wu, Sep 05 2016

Formula

a(n) = (Sum_{d | n} 0.d) * A276467(n).

A078268 Smallest integer which is an integer multiple of the number N obtained by placing the string "n" after a decimal point.

Original entry on oeis.org

1, 1, 3, 2, 1, 3, 7, 4, 9, 1, 11, 3, 13, 7, 3, 4, 17, 9, 19, 1, 21, 11, 23, 6, 1, 13, 27, 7, 29, 3, 31, 8, 33, 17, 7, 9, 37, 19, 39, 2, 41, 21, 43, 11, 9, 23, 47, 12, 49, 1, 51, 13, 53, 27, 11, 14, 57, 29, 59, 3, 61, 31, 63, 16, 13, 33, 67, 17, 69, 7, 71, 18, 73, 37, 3, 19, 77, 39, 79
Offset: 1

Views

Author

Amarnath Murthy, Nov 24 2002

Keywords

Comments

Numerator of n/10^k, where k is the number of digits in n. - Dean Hickerson, Mar 21 2003
a(p) = p if p is a prime other than 2 and 5.
Smallest integer m such that the concatenation of decimal representations of m and n is a multiple of n. - Reinhard Zumkeller, Mar 19 2003
a(n) = numerator of fraction a/b, where gcd(a, b) = 1, such that its decimal representation has the form 0.(n). Denominators in A078267: 10, 5, 10, 5, 2, 5, 10, 5, 10, 10, 100, ... Example: a(6) = 3; 3/5 = 0.6. - Jaroslav Krizek, Feb 05 2010
a(n) = n iff gcd(n,10) = 1. - Robert Israel, Jul 25 2014

Examples

			a(40)=2 since writing 40 after the decimal point gives 0.40 and 2 is the smallest integer multiple of 0.4.

Links

Crossrefs

Cf. A078267.

Programs

  • Maple
    a:= n -> numer(n/10^(1+ilog10(n))):
seq(a(n),n=1..100); # Robert Israel, Jul 25 2014
  • Mathematica
    si[n_]:=Module[{c=n/10^IntegerLength[n],m=1},While[!IntegerQ[c*m],m++]; c*m]; Array[si,80] (* Harvey P. Dale, Apr 06 2013 *)
Table[n/GCD[n, 10^(1 + Floor[Log10[n]])], {n, 79}] (* L. Edson Jeffery, Jul 25 2014 *)
  • PARI
    a(n) = numerator(n/10^(#Str(n))); \\ Michel Marcus, Mar 31 2019

Formula

a(n) = n *A078267(n)/10^A055642(n). - Jaroslav Krizek, Feb 05 2010
a(n) = n/A068822(n). - L. Edson Jeffery, Jul 25 2014

Extensions

Edited and extended by Henry Bottomley, Dec 08 2002
Incorrect formula removed by Jaroslav Krizek, Feb 05 2010

A276467 a(n) = denominator of Sum_{d|n} 0.d.

Original entry on oeis.org

10, 10, 5, 10, 5, 5, 5, 2, 10, 10, 100, 25, 100, 50, 20, 50, 100, 25, 100, 2, 100, 100, 100, 25, 20, 100, 100, 50, 100, 4, 100, 50, 25, 100, 20, 25, 100, 100, 25, 10, 100, 100, 100, 100, 5, 100, 100, 5, 100, 20, 25, 100, 100, 100, 50, 50, 25, 100, 100, 100
Offset: 1

Views

Author

Jaroslav Krizek, Sep 05 2016

Keywords

Comments

Here 0.d means the decimal fraction obtained by writing d after the decimal point, e.g., 0.12 = 12/100 = 3/25.
The first few values of Sum_{d|n} 0.d for n=1,2,.. are: 1/10, 3/10, 2/5, 7/10, 3/5, 6/5, 4/5, 3/2, 13/10, 9/10, 21/100, 43/25, ...
a(16450) = 1: 16450 is the only integer < 5*10^7 such that Sum_{d|n} 0.d is an integer; Sum_{d|16450} 0.d = 0.1 + 0.2 + 0.5 + 0.7 + 0.10 + 0.14 + 0.25 + 0.35 + 0.47 + 0.50 + 0.70 + 0.94 + 0.175 + 0.235 + 0.329 + 0.350 + 0.470 + 0.658 + 0.1175 + 0.1645 + 0.2350 + 0.3290 + 0.8225 + 0.16450 = 9; see A276465.
No other term like 16450 up to 10^9. - Michel Marcus, Mar 30 2019
No other term like 16450 up to 4*10^11. - Giovanni Resta, Apr 03 2019

Examples

			For n=12: Sum_{d|12} 0.d = 0.1 + 0.2 + 0.3 + 0.4 + 0.6 + 0.12 = 1.72 = 172/100 = 43/25; a(12) = 25.

Crossrefs

Cf. A276465, A276466 (numerators).
Cf. A078267 and A078268 (both for 0.d).

Programs

  • Magma
    [Denominator(&+[d / (10^(#Intseq(d))): d in Divisors(n)]): n in [1..1000]]
  • Mathematica
    Table[Denominator@ Total@ (#*1/10^(1 + Floor@ Log10@ #)) &@ Divisors@ n, {n, 60}] (* Michael De Vlieger, Sep 06 2016 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, d/10^(#Str(d)))); \\ Michel Marcus, Sep 05 2016

Formula

a(n) = A276466(n) / (Sum_{d|n} 0.d).

A172507 a(n) = numerator of fraction a/b, where gcd(a, b) = 1, whose decimal representation has the form (n).(n).

Original entry on oeis.org

11, 11, 33, 22, 11, 33, 77, 44, 99, 101, 1111, 303, 1313, 707, 303, 404, 1717, 909, 1919, 101, 2121, 1111, 2323, 606, 101, 1313, 2727, 707, 2929, 303
Offset: 1

Views

Author

Jaroslav Krizek, Feb 05 2010

Keywords

Comments

Sequence of denominators: 10, 25, 1000, 5000, 20000, 15625, 10000000, 50000000, ... Conjecture: this sequence is not equal to the sequence A078267.

Examples

			a(6) = 33; 33/5 = 6.6.

A307313 a(n) is the denominator of n/2^(length of the binary representation of n).

Original entry on oeis.org

2, 2, 4, 2, 8, 4, 8, 2, 16, 8, 16, 4, 16, 8, 16, 2, 32, 16, 32, 8, 32, 16, 32, 4, 32, 16, 32, 8, 32, 16, 32, 2, 64, 32, 64, 16, 64, 32, 64, 8, 64, 32, 64, 16, 64, 32, 64, 4, 64, 32, 64, 16, 64, 32, 64, 8, 64, 32, 64, 16, 64, 32, 64, 2, 128, 64, 128, 32, 128, 64
Offset: 1

Views

Author

Michel Marcus, Apr 02 2019

Keywords

Examples

			For n=1, 1 = 1_2,  a(1) = denominator(1/(2^1)) = denominator(1/2) = 2;
For n=2, 2 = 10_2, a(2) = denominator(2/(2^2)) = denominator(1/2) = 2;
For n=3, 3 = 11_2, a(3) = denominator(3/(2^2)) = denominator(3/4) = 4.

Crossrefs

Cf. A062383, A070939, A000265 (numerators), A078267 (analog in base 10).

Programs

  • PARI
    a(n) = denominator(n/(2^(#binary(n))));

Formula

a(n) = denominator(n/2^A070939(n)).
a(n) = denominator(n/A062383(n)).
a(n) = 2^A070940(n).
Showing 1-5 of 5 results.

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