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 1-3 of 3 results.

A078267 Smallest k such that k*N is an integer where N is obtained by placing the string "n" after a decimal point.

Original entry on oeis.org

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

Views

Author

Amarnath Murthy, Nov 24 2002

Keywords

Comments

From Jaroslav Krizek, Feb 05 2010: (Start)
a(n) is the denominator of fraction a/b, where gcd(a, b) = 1, such that its decimal representation has form 0.(n).
The numerators are in A078268. Example: a(6) = 5; 3/5 = 0.6.
(End)

Examples

			a(40) = 5 since 5*0.40 = 2 is an integer. a(1) = a(10) = 10.

Links

Crossrefs

Cf. A055642, A078268.

Programs

  • Mathematica
    Array[#2/GCD[#1, #2] & @@ {#, 10^IntegerLength[#]} &, 65] (* Michael De Vlieger, Oct 05 2021 *)
  • PARI
    a(n) = denominator(n/10^(#Str(n))); \\ Michel Marcus, Mar 31 2019
  • Python
    from math import gcd
def a(n): b = 10**len(str(n)); return b//gcd(n, b)
print([a(n) for n in range(1, 103)]) # Michael S. Branicky, Oct 05 2021

Formula

a(10^m) = 10, a(r*10^m) = a(r).
a(n) = (A078268(n)*10^A055642(n)) / n. [Jaroslav Krizek, Feb 05 2010]
a(n) = 10^A055642(n)/gcd(n, 10^A055642(n)). - Michael S. Branicky, Oct 05 2021

Extensions

Edited and extended by Henry Bottomley, Dec 08 2002

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).

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).
Showing 1-3 of 3 results.

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