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.

A323780 a(n) = denominator of Sum_{d|n} (tau(d)/sigma(d)) where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 3, 2, 21, 3, 2, 4, 105, 26, 9, 6, 7, 7, 12, 1, 3255, 9, 26, 10, 63, 8, 18, 12, 35, 93, 21, 65, 21, 15, 3, 16, 1085, 4, 27, 3, 91, 19, 6, 7, 315, 21, 8, 22, 9, 13, 36, 24, 2170, 76, 279, 3, 147, 27, 39, 9, 21, 20, 9, 30, 21, 31, 48, 104, 137795, 21, 12, 34
Offset: 1

Views

Author

Jaroslav Krizek, Feb 13 2019

Keywords

Comments

Sum_{d|n} (tau(d)/sigma(d)) >= 1 for all n >= 1.

Examples

			For n=4; Sum_{d|4} (tau(d)/sigma(d)) = (tau(1)/sigma(1))+(tau(2)/sigma(2))+(tau(4)/sigma(4)) = (1/1)+(2/3)+(3/7) = 44/21; a(4) = 21.

Crossrefs

Cf. A000005, A000203, A323779 (numerator), A323781.

Programs

  • Magma
    [Denominator(&+[NumberOfDivisors(d) / SumOfDivisors(d): d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Array[Denominator@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &, 67] (* Michael De Vlieger, Feb 15 2019 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, numdiv(d)/sigma(d))); \\ Michel Marcus, Feb 13 2019

Formula

a(p) = (p+1) / gcd(p+3, p+1) for p = primes p.
a(n) = 1 for numbers in A323781.

A324500 a(n) = denominator of Sum_{d|n} sigma(d)/tau(d) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

Original entry on oeis.org

1, 2, 1, 6, 1, 2, 1, 12, 3, 1, 1, 2, 1, 2, 1, 60, 1, 3, 1, 3, 1, 2, 1, 4, 3, 1, 3, 6, 1, 1, 1, 60, 1, 1, 1, 9, 1, 2, 1, 3, 1, 2, 1, 6, 3, 2, 1, 20, 1, 6, 1, 3, 1, 3, 1, 12, 1, 1, 1, 1, 1, 2, 3, 420, 1, 2, 1, 3, 1, 1, 1, 18, 1, 1, 1, 6, 1, 1, 1, 15, 15, 1, 1, 2
Offset: 1

Views

Author

Jaroslav Krizek, Mar 02 2019

Keywords

Comments

Sum_{d|n} sigma(d)/tau(d) > 1 for all n > 1.
Sum_{d|n} sigma(d)/tau(d) = n only for numbers n = 1, 3, 10 and 30.

Examples

			Sum_{d|n} sigma(d)/tau(d) for n >= 1: 1, 5/2, 3, 29/6, 4, 15/2, 5, 103/12, 22/3, 10, 7, 29/2, 8, 25/2, 12, 887/60, ...
For n=4; Sum_{d|4} sigma(d)/tau(d) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(4)/tau(4) = 1/1 + 3/2 + 7/3 = 29/6;  a(4) = 6.

Links

Crossrefs

Cf. A000005, A000203, A323779, A323780, A323781, A324499 (numerators), A306639.

Programs

  • Magma
    [Denominator(&+[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Table[Denominator[Sum[DivisorSigma[1, k]/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 100}] (* G. C. Greubel, Mar 04 2019 *)
  • PARI
    a(n) = denominator(sumdiv(n, d, sigma(d)/numdiv(d))); \\ Michel Marcus, Mar 03 2019
  • Sage
    [sum(sigma(k,1)/sigma(k,0) for k in n.divisors() ).denominator() for n in (1..100)] # G. C. Greubel, Mar 04 2019

Formula

a(p) = 1 for odd primes p.
a(n) = 1 for numbers in A306639.

A324499 a(n) = numerator of Sum_{d|n} sigma(d)/tau(d) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of the divisors of k (A000005).

Original entry on oeis.org

1, 5, 3, 29, 4, 15, 5, 103, 22, 10, 7, 29, 8, 25, 12, 887, 10, 55, 11, 58, 15, 35, 13, 103, 43, 20, 52, 145, 16, 30, 17, 1517, 21, 25, 20, 319, 20, 55, 24, 103, 22, 75, 23, 203, 88, 65, 25, 887, 24, 215, 30, 116, 28, 130, 28, 515, 33, 40, 31, 58, 32, 85, 110
Offset: 1

Views

Author

Jaroslav Krizek, Mar 02 2019

Keywords

Comments

Sum_{d|n} sigma(d)/tau(d) > 1 for all n > 1.
Sum_{d|n} sigma(d)/tau(d) = n only for numbers n = 1, 3, 10 and 30.

Examples

			Sum_{d|n} sigma(d)/tau(d) for n >= 1: 1, 5/2, 3, 29/6, 4, 15/2, 5, 103/12, 22/3, 10, 7, 29/2, 8, 25/2, 12, 887/60, ...
For n=4; Sum_{d|4} sigma(d)/tau(d) = sigma(1)/tau(1) + sigma(2)/tau(2) + sigma(4)/tau(4) = 1/1 + 3/2 + 7/3 = 29/6; a(4) = 29.

Crossrefs

Cf. A000005, A000203, A323779, A323780, A323781, A324500 (denominators).

Programs

  • Magma
    [Numerator(&+[SumOfDivisors(d) / NumberOfDivisors(d): d in Divisors(n)]): n in [1..100]]
  • Mathematica
    Table[Numerator[Sum[DivisorSigma[1, k]/DivisorSigma[0, k], {k, Divisors[n]}]], {n, 1, 100}] (* G. C. Greubel, Mar 04 2019 *)
  • PARI
    a(n) = numerator(sumdiv(n, d, sigma(d)/numdiv(d))); \\ Michel Marcus, Mar 03 2019
  • Sage
    [sum(sigma(k,1)/sigma(k,0) for k in n.divisors() ).numerator() for n in (1..100)] # G. C. Greubel, Mar 04 2019

A323781 Numbers m such that Sum_{d|m} (tau(d)/sigma(d)) is an integer h where tau(k) = the number of the divisors of k (A000005) and sigma(k) = the sum of the divisors of k (A000203).

Original entry on oeis.org

1, 15, 429, 609, 6003, 9156, 20943, 75579, 90252, 93849, 115773, 331359, 631764, 744993, 817191, 837655, 925083, 1130766, 1141191, 2349087, 2491740, 2512965, 3040728, 3266253, 3796143, 4314891, 4365231, 5025930, 5294340, 6135624, 6629271, 7210671, 10906175
Offset: 1

Views

Author

Jaroslav Krizek, Feb 16 2019

Keywords

Comments

Sum_{d|n} (tau(d)/sigma(d)) > 1 for all n > 2.
Corresponding values of integers h: 1, 2, 2, 2, 2, 4, 2, 2, 4, 2, 2, 2, 5, 2, 2, 2, 2, 4, 2, 2, 5, 3, 4, 2, 2, 2, 2, 5, 5, 5, 2, 2, 2, ...
The smallest number m such that Sum_{d|m} (tau(d)/sigma(d)) is an integer h for h >= 1: 1, 15, 2512965, 9156, 631764, ...

Examples

			15 is a term because Sum_{d|15} (tau(d)/sigma(d)) = tau(1)/sigma(1) + tau(3)/sigma(3) + tau(5)/sigma(5) + tau(15)/sigma(15) = 1/1 + 2/4 + 2/6 + 4/24 = 2 (integer).

Crossrefs

Cf. A000005, A000203, A323779, A323780.

Programs

  • Magma
    [n: n in [1..1000000] | Denominator(&+[NumberOfDivisors(d) / SumOfDivisors(d): d in Divisors(n)]) eq 1]
  • Mathematica
    Select[Range[10^5], IntegerQ@ DivisorSum[#, Divide @@ DivisorSigma[{0, 1}, #] &] &] (* Michael De Vlieger, Feb 17 2019 *)
  • PARI
    isok(n) = !frac(sumdiv(n, d, numdiv(d)/sigma(d))); \\ Michel Marcus, Feb 16 2019

Formula

A323780(a(n)) = 1.
Showing 1-4 of 4 results.

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