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).

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).

A276699 Numbers n such that Sum_{q|n} 0.q is an integer where q ranges over the aliquot parts of n.

Original entry on oeis.org

14, 297, 869, 1241, 1349, 1541, 1769, 1829, 1961, 2021, 82091, 88931, 98171, 100739, 105779, 111899, 116651, 122411, 125771, 130139, 135419, 139499, 150971, 152771, 157979, 158819, 165251, 169739, 173939, 174611, 177851, 182051, 183731, 188339, 189731, 193091
Offset: 1

Views

Author

Paolo P. Lava, Sep 15 2016

Keywords

Comments

Mainly numbers ending in 1 or 9.
So far Sum{q | n} 0.q is generally equal to 1 apart from 297 with 3 and 235569 with 6.

Examples

			Aliquot parts of 14 are 1, 2, 7 and 0.1 + 0.2 + 0.7 = 1;
Aliquot parts of 297 are 1, 3, 9, 11, 27, 33, 99 and 0.1 + 0.3 + 0.9 + 0.11 + 0.27 + 0.33 + 0.99 = 3;

Links

Crossrefs

Cf. A276465, A276655, A276700.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a,n;
for n from 2 to q do a:=sort([op(divisors(n))]);
if type(add(a[k]/10^(ilog10(a[k])+1),k=1..nops(a)-1),integer)
then print(n); fi; od; end: P(10^9);

A276700 Numbers n such that Sum_{p|n} 0.p is an integer where p ranges over the prime divisors, with multiplicity, of n.

Original entry on oeis.org

1, 21, 25, 30, 32, 36, 392, 441, 525, 560, 625, 630, 672, 750, 756, 800, 900, 960, 979, 1024, 1080, 1152, 1215, 1296, 1411, 1458, 1463, 1547, 1742, 1947, 2059, 2090, 2210, 2318, 2405, 2419, 2444, 2491, 2508, 2552, 2652, 2703, 2871, 2886, 2924, 2945, 3116, 3128
Offset: 1

Views

Author

Paolo P. Lava, Sep 15 2016

Keywords

Examples

			21 = 3 * 7 and 0.3 + 0.7 = 1;
3128 = 2^3 * 17 * 23 and 3*0.2 + 0.17 + 0.23 = 1.

Links

Crossrefs

Cf. A276465, A276655, A276699.

Programs

  • Maple
    with(numtheory): P:= proc(q) local a,k,n;
for n from 1 to q do a:=ifactors(n)[2];
if type(add(a[k][2]*a[k][1]/10^(ilog10(a[k][1])+1),k=1..nops(a)),integer)
then print(n); fi; od; end: P(10^9);
  • Mathematica
    paiQ[n_]:=IntegerQ[Total[#/10^IntegerLength[#]&/@Flatten[Table[#[[1]], #[[2]]]&/@ FactorInteger[n]]]]; Join[{1},Select[Range[3200],paiQ]] (* Harvey P. Dale, Apr 02 2020 *)
Showing 1-4 of 4 results.

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