A009205 -id:A009205 - cp's OEIS frontend

A009194 a(n) = gcd(n, sigma(n)).

1, 1, 1, 1, 1, 6, 1, 1, 1, 2, 1, 4, 1, 2, 3, 1, 1, 3, 1, 2, 1, 2, 1, 12, 1, 2, 1, 28, 1, 6, 1, 1, 3, 2, 1, 1, 1, 2, 1, 10, 1, 6, 1, 4, 3, 2, 1, 4, 1, 1, 3, 2, 1, 6, 1, 8, 1, 2, 1, 12, 1, 2, 1, 1, 1, 6, 1, 2, 3, 2, 1, 3, 1, 2, 1, 4, 1, 6, 1, 2, 1, 2, 1, 28, 1, 2, 3, 4, 1, 18, 7, 4, 1, 2, 5, 12, 1, 1, 3, 1, 1, 6, 1, 2

Offset: 1

David W. Wilson

nonn

LCM of common divisors of n and sigma(n). It equals n if n is multiply perfect (A007691). - Labos Elemer, Aug 14 2002

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
P. Pollack, On the greatest common divisor of a number and its sum of divisors, Michigan Math. J. Volume 60, Issue 1 (2011), 199-214.

Cf. A000203, A003624, A007691, A014567, A017666, A063906, A069059, A073802, A082062, A179931, A205523, A216793 (positions of records), A234367, A249917.

Cf. also A009191, A009205, A009242, A274382, A286591, A286594.

a009194 n = gcd (a000203 n) n  -- Reinhard Zumkeller, Mar 23 2013

Table[GCD[n,DivisorSigma[1,n]],{n,110}] (* Harvey P. Dale, Aug 23 2015 *)

a(n) = gcd(n, sigma(n)); \\ Michel Marcus, Oct 23 2013

A000005(a(n)) = A073802(n). - Reinhard Zumkeller, Mar 12 2010
A006530(a(n)) = A082062(n). - Reinhard Zumkeller, Jul 10 2011
a(A014567(n)) = 1; A069059(a(n)) > 1. - Reinhard Zumkeller, Mar 23 2013
a(n) = n/A017666(n). - Antti Karttunen, May 22 2017

A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

Original entry on oeis.org

1, 2, 1, 1, 1, 2, 1, 4, 3, 2, 1, 6, 1, 2, 1, 1, 1, 6, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 9, 1, 2, 1, 8, 1, 2, 1, 2, 3, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 8, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 2, 1, 2, 1, 12, 1, 2, 3, 2, 1, 2, 1, 10, 1, 2, 1, 12, 1, 2, 1, 8, 1, 6, 1, 2, 1, 2, 1, 12, 1, 2, 3, 1, 1, 2, 1, 8, 1

Offset: 1

David W. Wilson

nonn

a(A046642(n)) = 1.
First occurrence of k: 1, 2, 9, 8, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 384, .... Conjecture: each k is present. - Robert G. Wilson v, Mar 27 2013
Conjecture is true. See David A. Corneth's comment in A324553. - Antti Karttunen, Mar 06 2019

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from T. D. Noe)

Cf. A000005, A009194, A009195, A009205, A009213, A009230, A049820, A125168, A138010, A286540, A303781, A318459, A319337, A322979, A322980, A323073.

Cf. A046642 (positions of ones), A324553 (position of the first occurrence of each n).

a009191 n = gcd n $ a000005 n
-- Reinhard Zumkeller, May 09 2013, Aug 14 2011

f[n_] := GCD[n, DivisorSigma[0, n]]; Array[f, 105] (* Robert G. Wilson v, Mar 27 2013 *)

a(n)=gcd(numdiv(n),n) \\ Charles R Greathouse IV, Mar 26 2013

a(n) = gcd(n, A000005(n)) = gcd(n, A049820(n)). - Antti Karttunen, Sep 25 2018

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

A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 2, 1, 1, 1, 4, 1, 4, 3, 4, 1, 6, 1, 4, 1, 1, 1, 6, 1, 2, 1, 4, 1, 8, 1, 4, 1, 2, 1, 8, 1, 2, 1, 4, 1, 9, 1, 4, 1, 8, 1, 8, 1, 2, 3, 4, 1, 2, 1, 2, 1, 2, 1, 8, 1, 8, 1, 4, 1, 12, 1, 4, 3, 1, 1, 8, 1, 2, 1, 8, 1, 12, 1, 4, 3, 2, 1, 8, 1, 10, 1, 4, 1, 12, 1, 4

Offset: 1

Jaroslav Krizek, Mar 04 2019

nonn

Sequence of the smallest numbers k such that a(k) = n: 1, 2, 9, 6, 400, 12, 3136, 24, 36, 80, 123904, 60, 692224, 448, 2025, 120, ...

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A007955, A009205, A046642.

[GCD(NumberOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = gcd(numdiv(n), vecprod(divisors(n))); \\ Michel Marcus, Mar 04 2019

a(n) = 1 for numbers in A046642.

a(n) = tau(n) for numbers in A120736.

A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 2, 1, 4, 1, 1, 1, 3, 1, 2, 1, 4, 1, 4, 1, 2, 1, 2, 1, 8, 1, 1, 1, 2, 1, 1, 1, 4, 1, 2, 1, 8, 1, 2, 3, 4, 1, 2, 1, 1, 1, 2, 1, 8, 1, 8, 1, 2, 1, 12, 1, 4, 1, 1, 1, 8, 1, 2, 1, 8, 1, 3, 1, 2, 1, 2, 1, 8, 1, 2, 1, 2, 1, 4, 1, 4, 1, 4, 1, 6, 1, 2, 1, 4, 1, 12, 1, 1, 3, 1, 1, 8, 1, 2, 1

Offset: 1

Jaroslav Krizek, Aug 01 2020

nonn

a(n) = tau(n) for numbers n: 1, 6, 14, 22, 30, 38, 42, 46, 54, 56, 60, 62, 66, 70, 78, 86, 94, 96, 102, ...

			a(6) = gcd(tau(6), sigma(6), pod(6)) = gcd(4, 12, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A009205 (gcd(tau(n), sigma(n))), A306671 (gcd(tau(n), pod(n))), A306682 (gcd(sigma(n), pod(n))).
Cf. A000005 (tau(n)), A000203 (sigma(n)), A007955 (pod(n)), A336723 (lcm(tau(n), sigma(n), pod(n))).
Cf. A277521 (numbers k such that a(k) = tau(k) and simultaneously A336723(k) = pod(k)).

[GCD([#Divisors(n), &+Divisors(n), &*Divisors(n)]): n in [1..100]];

a[n_] := GCD @@ {(d = DivisorSigma[0,n]), DivisorSigma[1, n], n^(d/2)}; Array[a, 100] (* Amiram Eldar, Aug 01 2020 *)

A007955(n) = if(issquare(n, &n), n^numdiv(n^2), n^(numdiv(n)/2)); \\ From A007955
A336722(n) = gcd(A007955(n), gcd(numdiv(n), sigma(n))); \\ Antti Karttunen, Aug 10 2020

a(p) = 1 for p = primes (A000040).

a(n) = gcd(A007955(n), A009205(n)). - Antti Karttunen, Aug 10 2020

Data section extended up to a(105) by Antti Karttunen, Aug 10 2020

A009278 a(n) = lcm(d(n), sigma(n)).

Original entry on oeis.org

1, 6, 4, 21, 6, 12, 8, 60, 39, 36, 12, 84, 14, 24, 24, 155, 18, 78, 20, 42, 32, 36, 24, 120, 93, 84, 40, 168, 30, 72, 32, 126, 48, 108, 48, 819, 38, 60, 56, 360, 42, 96, 44, 84, 78, 72, 48, 620, 57, 186, 72, 294, 54, 120, 72, 120, 80, 180, 60, 168, 62, 96, 312, 889, 84, 144, 68

Offset: 1

David W. Wilson

nonn

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000005, A000203, A009205, A009286, A064840.

Table[LCM[Length[Divisors[n]], DivisorSigma[1, n]], {n, 1, 100}] (* Stefan Steinerberger, Apr 13 2006 *)

A009278(n) = lcm(numdiv(n), sigma(n)); \\ Antti Karttunen, Sep 10 2017

a(n) = A064840(n)/A009205(n). - Amiram Eldar, Jan 31 2025

A057022 a(n) = floor((sum of divisors of n) / (number of divisors of n)), or floor(sigma_1(n)/sigma_0(n)).

Original entry on oeis.org

1, 1, 2, 2, 3, 3, 4, 3, 4, 4, 6, 4, 7, 6, 6, 6, 9, 6, 10, 7, 8, 9, 12, 7, 10, 10, 10, 9, 15, 9, 16, 10, 12, 13, 12, 10, 19, 15, 14, 11, 21, 12, 22, 14, 13, 18, 24, 12, 19, 15, 18, 16, 27, 15, 18, 15, 20, 22, 30, 14, 31, 24, 17, 18, 21, 18, 34, 21, 24, 18

Offset: 1

Henry Bottomley, Jul 21 2000

nonn

Floor of mean of divisors of n. - Jon E. Schoenfield, Dec 24 2016

			a(4)=2 since the 3 divisors of 4 are 1, 2 and 4 and floor((1 + 2 + 4)/3) = floor(7/3) = 2.

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

Cf. A000005, A000203, A009205, A054025, A057020, A057021.

Cf. A003601.

a057022 n = a000203 n `div` a000005 n
-- Reinhard Zumkeller, Jan 06 2012

Floor[Table[Total[Divisors[n]]/Length[Divisors[n]],{n,20}]] (* Daniel Jolly, Nov 15 2014 *)
Table[Floor[DivisorSigma[1,n]/DivisorSigma[0,n]],{n,70}] (* Harvey P. Dale, Jan 14 2015 *)

a(n) = sigma(n)\numdiv(n); \\ Michel Marcus, Nov 15 2014

a(n) = (A000203(n) - A054025(n))/A000005(n).

A306682 a(n) = gcd(sigma(n), pod(n)) where sigma(k) = the sum of the divisors of k (A000203) and pod(k) = the product of the divisors of k (A007955).

Original entry on oeis.org

1, 1, 1, 1, 1, 12, 1, 1, 1, 2, 1, 4, 1, 4, 3, 1, 1, 3, 1, 2, 1, 4, 1, 12, 1, 2, 1, 56, 1, 72, 1, 1, 3, 2, 1, 1, 1, 4, 1, 10, 1, 48, 1, 4, 3, 4, 1, 4, 1, 1, 9, 2, 1, 24, 1, 8, 1, 2, 1, 24, 1, 4, 1, 1, 1, 144, 1, 2, 3, 16, 1, 3, 1, 2, 1, 4, 1, 24, 1, 2, 1, 2, 1

Offset: 1

Jaroslav Krizek, Mar 05 2019

nonn

See A324527(n) = the smallest numbers k such that a(k) = n.

			For n=6: a(6) = gcd(tau(6), pod(6)) = gcd(4, 36) = 4.

Antti Karttunen, Table of n, a(n) for n = 1..16384
Antti Karttunen, Data supplement: n, a(n) computed for n = 1..65537

Cf. A000005, A000203, A007955, A009205, A046642, A120736, A324526, A324527.

[GCD(SumOfDivisors(n), &*[d: d in Divisors(n)]): n in [1.. 100]]

a(n) = my(d=divisors(n)); gcd(vecsum(d), vecprod(d)); \\ Michel Marcus, Mar 05 2019

a(n) = 1 for numbers in A014567.

a(n) = tau(n) for numbers in A324526.

A336856 Prime-shifted analog of gcd(d(n), sigma(n)): a(n) = gcd(A000005(n), A003973(n)).

Original entry on oeis.org

1, 2, 2, 1, 2, 4, 2, 4, 1, 4, 2, 6, 2, 4, 4, 1, 2, 2, 2, 2, 4, 4, 2, 8, 3, 4, 4, 6, 2, 8, 2, 2, 4, 4, 4, 1, 2, 4, 4, 8, 2, 8, 2, 2, 2, 4, 2, 2, 1, 6, 4, 6, 2, 8, 4, 8, 4, 4, 2, 12, 2, 4, 6, 1, 4, 8, 2, 2, 4, 8, 2, 4, 2, 4, 6, 6, 4, 8, 2, 2, 1, 4, 2, 12, 4, 4, 4, 8, 2, 4, 4, 6, 4, 4, 4, 12, 2, 2, 2, 3, 2, 8, 2, 8, 8

Offset: 1

Antti Karttunen, Aug 12 2020

nonn

Cf. A000005, A003961, A003973, A009205, A336837, A336838, A336839, A336840, A336841.

A003973(n) = { my(f = factor(n)); for(i=1, #f~, f[i, 1] = nextprime(f[i, 1]+1)); sigma(factorback(f)); };
A336856(n) = gcd(numdiv(n), A003973(n));

a(n) = A009205(A003961(n)).
a(n) = gcd(A000005(n), A003973(n)) = gcd(A000005(n), A336841(n)).
a(n) = gcd(A000005(n), 2*A336840(n)).
a(n) = A003973(n) / A336838(n) = A000005(n) / A336839(n).
For n > 1, a(n) = A336841(n) / A336837(n).
For all primes p, and n >= 0, a(p^((2^n)-1)) = 2^n.

cp's OEIS Frontend

A009194 a(n) = gcd(n, sigma(n)).

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A009191 a(n) = gcd(n, d(n)), where d(n) is the number of divisors of n (A000005).

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

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

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A306671 a(n) = gcd(tau(n), pod(n)) where tau(k) = the number of the divisors of k (A000005) and pod(k) = the product of the divisors of k (A007955).

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A336722 a(n) = gcd(tau(n), sigma(n), pod(n)) where tau(k) is the number of divisors of k (A000005), sigma(k) is the sum of divisors of k (A000203) and pod(k) is the product of divisors of k (A007955).

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