A057022 -id:A057022 - cp's OEIS frontend

A348182 a(1) = 1; for n >= 2, a(n) = 1 + a(A057022(n)).

1, 2, 3, 3, 4, 4, 4, 4, 4, 4, 5, 4, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 5, 6, 5, 6, 5, 5, 6, 5, 5, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 5, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 7, 6, 6, 7, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 6, 6, 6, 6, 7, 6, 6, 6, 7, 6, 7, 6

Offset: 1

Ctibor O. Zizka, Oct 05 2021

nonn

Number of steps needed to reach one when starting from k = n and repeatedly applying the map that replaces k by A057022(k). First maximal values for n = 1,2,3,5,11,29,61, .. which, except 1, are all primes (empirical result).

			a(5) = 1 + a(3) = 1 + 1 + a(2) = 1 + 1 + 1 + a(1) = 1 + 1 + 1 + 1 = 4.

Cf. A057022.

a[1] = 1; a[n_] := a[n] = 1 + a[Floor[DivisorSigma[1, n]/DivisorSigma[0, n]]]; Array[a, 100] (* Amiram Eldar, Oct 05 2021 *)

A057021 Denominator of (sum of divisors of n / number of divisors of n).

Original entry on oeis.org

1, 2, 1, 3, 1, 1, 1, 4, 3, 2, 1, 3, 1, 1, 1, 5, 1, 2, 1, 1, 1, 1, 1, 2, 3, 2, 1, 3, 1, 1, 1, 2, 1, 2, 1, 9, 1, 1, 1, 4, 1, 1, 1, 1, 1, 1, 1, 5, 1, 2, 1, 3, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1, 1, 1

Offset: 1

Henry Bottomley, Jul 21 2000

a(n) = 1 when n is listed in A003601, a(n) > 1 when n is listed in A049642. - Alonso del Arte, Jan 31 2006

a(A069081(n)) = 2. - Bernard Schott, Sep 19 2019

			a(12)=3 since the 6 divisors of 12 are 1, 2, 3, 4, 6 and 12 and 1+2+3+4+6+12=28 and 28/6=14/3.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A009205, A054025, A057020 (numerator), A057022, A069081.

import Data.Ratio ((%), denominator)
a057021 n = denominator $ a000203 n % a000005 n
-- Reinhard Zumkeller, Jan 06 2012

[Denominator(SumOfDivisors(n)/#Divisors(n)):n in [1..100]]; // Marius A. Burtea, Sep 08 2019

with(numtheory): seq(denom(sigma(n)/tau(n)), n=1..70) ; # Zerinvary Lajos, Jun 04 2008

Denominator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* Alonso del Arte, Feb 24 2006 *)

a(n) = denominator(sigma(n)/numdiv(n)); \\ Michel Marcus, Apr 12 2016

[denominator(sigma(n, 1)/sigma(n, 0)) for n in range(1, 71)] # Stefano Spezia, Jul 18 2025

a(n) = A057020(n)*A000005(n)/A000203(n) = A000005(n)/A009205(n).

A057020 Numerator of (sum of divisors of n / number of divisors of n).

Original entry on oeis.org

1, 3, 2, 7, 3, 3, 4, 15, 13, 9, 6, 14, 7, 6, 6, 31, 9, 13, 10, 7, 8, 9, 12, 15, 31, 21, 10, 28, 15, 9, 16, 21, 12, 27, 12, 91, 19, 15, 14, 45, 21, 12, 22, 14, 13, 18, 24, 62, 19, 31, 18, 49, 27, 15, 18, 15, 20, 45, 30, 14, 31, 24, 52, 127, 21, 18, 34, 21, 24, 18

Offset: 1

Henry Bottomley, Jul 21 2000

Numerator of arithmetic mean of the divisors of n. - Jaroslav Krizek, Apr 26 2010

The average order of a(n)/A057021(n) is asymptotic to n/sqrt(log(n)); see the Bateman et al. link or the Sutantyo link. - Charles R Greathouse IV, May 17 2012

			a(12) = 14 since the 6 factors of 12 are 1, 2, 3, 4, 6 and 12 and 1 + 2 + 3 + 4 + 6 + 12 = 28 and 28/6 = 14/3.

V. I. Arnold, Dynamics, Statistics, and Projective Geometry of Galois Fields, Cambridge University Press, Cambridge, 2011, p. 78.

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
V. Arnold, Number-theoretical turbulence in Fermat-Euler arithmetics and large Young diagrams geometry statistics, Journal of Mathem. Fluid Mechanics 7 (2005), pp. S4-S50.
Paul T. Bateman, Paul Erdős, Carl Pomerance, and Ernst G. Straus, The arithmetic mean of the divisors of an integer in Analytic Number Theory (1980), pp. 197-220.
Marcin Mazur and Bogdan V. Petrenko, Representations of analytic functions as infinite products and their application to numerical computations, The Ramanujan Journal, Vol. 34, No. 1 (2014), pp. 129-141; arXiv preprint, arXiv:1202.1335 [math.NT], 2012.
Daniel Sutantyo, Elementary and Analytic Methods in Number Theory, M.S. thesis (Macquarie University, 2007), chapter 3. [Wayback Machine link]

Cf. A000005, A000203, A009205, A054025, A057021 (denominator), A057022, A308051.

import Data.Ratio ((%), numerator)
a057020 n = numerator $ a000203 n % a000005 n
-- Reinhard Zumkeller, Jan 06 2012

with(numtheory): seq(numer(sigma(n)/tau(n)), n=1..70) ; # Zerinvary Lajos, Jun 04 2008

Numerator[Table[(Plus @@ Divisors[n])/Length[Divisors[n]], {n, 70}]] (* Alonso del Arte, Feb 24 2006 *)
Table[Numerator[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,100}] (* Harvey P. Dale, Dec 19 2023 *)

a(n)=numerator(sigma(n)/numdiv(n)) \\ Charles R Greathouse IV, May 17 2012

[numerator(sigma(n, 1)/sigma(n, 0)) for n in range(1, 71)] # Stefano Spezia, Jul 18 2025

a(n) = A057021(n) * A000203(n)/A000005(n) = A000203(n)/A009205(n) = (A057022(n) + A054025(n)/A000005(n)) * A057021(n).

Sum_{k=1..n} a(k)/A057021(k) ~ c * n^2 /sqrt(log(n)), where c = A308051. - Amiram Eldar, Apr 15 2025

A324990 a(n) = the smallest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

Original entry on oeis.org

1, 3, 5, 7, 0, 11, 13, 21, 17, 19, 40, 23, 34, 39, 29, 31, 63, 46, 37, 57, 41, 43, 76, 47, 0, 99, 53, 74, 0, 59, 61, 93, 86, 67, 116, 71, 73, 111, 125, 79, 175, 83, 171, 121, 89, 122, 0, 141, 97, 0, 101, 103, 0, 107, 109, 188, 113, 250, 0, 158, 169, 183, 166

Offset: 1

Jaroslav Krizek, Mar 23 2019

Floor(sigma(n)/tau(n)) = floor(A000203(n)/A000005(n)) = A057022(n) for n >= 1.
Odd primes are terms.
a(n) = 0 for numbers n =  5, 25, 29, 47, 50, 53, 59, 83, 89, ...

			For n = 4; number 7 is the smallest number k with floor(sigma(k)/tau(k)) = 4; floor(sigma(7)/tau(7)) = floor(8/2) = 4.

Robert Israel, Table of n, a(n) for n = 1..2000

Cf. A000005, A000203, A057022, A162538, A324991.

Magma
```
Ax:=func; [Ax(n): n in[1..80]]
```

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for k from 1 to N^2 do
  v:= floor(numtheory:-sigma(k)/numtheory:-tau(k));
  if v <= N and V[v]=0 then V[v]:= k fi
od:
convert(V,list); # Robert Israel, Sep 13 2020

A324991 a(n) = the largest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

Original entry on oeis.org

2, 4, 8, 12, 0, 18, 24, 21, 30, 36, 40, 48, 45, 60, 56, 72, 63, 84, 90, 75, 96, 120, 108, 112, 0, 144, 110, 140, 0, 180, 160, 156, 136, 67, 116, 210, 240, 200, 198, 252, 175, 224, 208, 225, 288, 228, 0, 360, 336, 0, 172, 315, 0, 330, 272, 420, 294, 306, 0, 396

Offset: 1

Jaroslav Krizek, Mar 23 2019

nonn

Floor(sigma(n)/tau(n)) = floor(A000203(n)/A000005(n)) = A057022(n) for n >= 1.

a(n) = 0 for numbers n = 5, 25, 29, 47, 50, 53, 59, 83, 89, ...

			For n = 4; number 12 is the largest number k with floor(sigma(k)/tau(k)) = 4; floor(sigma(12)/tau(12)) = floor(28/6) = 4.

Cf. A000005, A000203, A057022, A162538, A324990.

[Max([n: n in[1..10^5] | Floor(SumOfDivisors(n)/ NumberOfDivisors(n)) eq k]): k in [1..4]] cat [0] cat [Max([n: n in[1..10^5] | Floor(SumOfDivisors(n)/ NumberOfDivisors(n)) eq k]): k in [6..24]];

cp's OEIS Frontend

A348182 a(1) = 1; for n >= 2, a(n) = 1 + a(A057022(n)).

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Mathematica

A057021 Denominator of (sum of divisors of n / number of divisors of n).

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Haskell

Magma

Maple

Mathematica

PARI

SageMath

Formula

A057020 Numerator of (sum of divisors of n / number of divisors of n).

Views

Author

Keywords

Comments

Examples

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

SageMath

Formula

A324990 a(n) = the smallest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Magma

Maple

A324991 a(n) = the largest number k such that floor(sigma(k)/tau(k)) = n, or 0 if no such number k exists.

Views

Author

Keywords

Comments

Examples

Crossrefs

Programs

Magma