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 1-4 of 4 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).

A276465 Divisors of 16450.

Original entry on oeis.org

1, 2, 5, 7, 10, 14, 25, 35, 47, 50, 70, 94, 175, 235, 329, 350, 470, 658, 1175, 1645, 2350, 3290, 8225, 16450
Offset: 1

Views

Author

Jaroslav Krizek, Sep 04 2016

Keywords

Comments

Conjecture: 16450 is the only number such that Sum_{d | n} 0.d is an integer, where 0.d means the decimal fraction of divisors d of n obtained by writing d after the decimal point:
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.
There are no other numbers with this property <= 5*10^7.

Crossrefs

Subsequences include A018262, A018270, A018319, A018406, A018472, and A018577.
Cf. A276466, A276467.

Programs

  • Magma
    Divisors(16450)
  • Mathematica
    Divisors@ 16450 (* generates sequence *)
Total@(#*1/10^(1 + Floor@ Log10@ #)) &@ Divisors@ 16450 (* illustrates comment, Michael De Vlieger, Sep 04 2016 *)
  • PARI
    divisors(16450) \\ Michel Marcus, Sep 04 2016

Formula

A276467(16450) = 1.

A276479 a(n) = floor(Sum_{d|n} 0.d).

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 1, 0, 0, 1, 0, 1, 1, 1, 0, 2, 0, 1, 1, 0, 0, 2, 0, 0, 1, 1, 0, 2, 0, 1, 0, 0, 1, 3, 0, 0, 0, 2, 0, 2, 0, 1, 2, 0, 0, 3, 1, 1, 1, 1, 0, 3, 1, 3, 1, 1, 0, 3, 0, 1, 2, 2, 1, 2, 0, 1, 1, 2, 0, 4, 0, 1, 2, 2, 1, 2, 0, 3, 2, 1, 0, 4, 1, 1, 1
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 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, ...
See A276480(n) = the smallest number k such that floor(Sum_{d|k} 0.d) = n.

Examples

			For n=12: a(12) = floor(Sum_{d|12} 0.d) = floor(0.1 + 0.2 + 0.3 + 0.4 + 0.6 + 0.12 = 0.72) = floor(172/100) = floor(43/25)  = 1.

Links

Crossrefs

Cf. A276466, A276467, A276480.

Programs

  • Magma
    [Floor(&+[d / (10^(#Intseq(d))): d in Divisors(n)]): n in [1..1000]];
  • Mathematica
    Table[Floor@ Total@ (#*1/10^(1 + Floor@ Log10@ #)) &@ Divisors@ n, {n, 120}] (* Michael De Vlieger, Sep 06 2016 *)
  • PARI
    a(n) = floor(sumdiv(n, d, d/10^(#Str(d)))); \\ Michel Marcus, Sep 05 2016
  • Python
    from fractions import Fraction
from sympy import divisors
def A276479(n):
    return sum(Fraction(d,10**len(str(d))) for d in divisors(n))._floor_() # Chai Wah Wu, Sep 08 2016

Formula

a(n) = floor(A276466(n)/A276467(n)).

A276480 a(n) = the smallest number k such that floor(Sum_{d|k} 0.d) = n.

Original entry on oeis.org

1, 6, 18, 36, 72, 120, 168, 288, 420, 360, 792, 720, 1512, 1260, 1440, 3240, 4032, 2880, 2520, 3960, 5544, 6720, 5040, 10920, 7560, 14400, 10080, 13860, 15840, 15120, 18480, 20160, 37440, 25200, 46800, 30240, 36960, 32760, 27720, 71280, 50400, 69300, 60480
Offset: 0

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.
a(n) = the smallest number k such that floor (A276466(k)/A276467(k)) = n.
The first few values of Sum_{d|n} 0.d 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, ...

Links

Crossrefs

Cf. A276466, A276467, A276479.

Programs

  • Mathematica
    Table[k = 1; While[Floor@ Total@ (#*1/10^(1 + Floor@ Log10@ #)) != n &@ Divisors@ k, k++]; k, {n, 0, 40}] (* Michael De Vlieger, Sep 06 2016 *)
  • PARI
    a(n) = {k = 1; while(floor(sumdiv(k, d, d/10^(#Str(d)))) != n, k++); k; } \\ Michel Marcus, Sep 05 2016
Showing 1-4 of 4 results.

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