A216793 -id:A216793 - 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

A245778 Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

Original entry on oeis.org

1, 672, 4680, 30240, 435708, 23569920, 45532800, 4138364160, 14182439040, 53798734080, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, Aug 01 2014

Numbers n such that A245776(n) / A245777(n) =  (n / A000005(n) - A000203(n) / n) is an integer.
Sequence of integers k(n): 0, 25, 94, 311, 4031, 73652, 118571, …
Conjecture: subsequence of A216793.
Refactorable multiply-perfect numbers (A245782) are members of this sequence.
a(14) > 10^13. - Giovanni Resta, Jul 13 2015
The numbers 13661860101120 and 740344994887680 are also terms. - Giovanni Resta, Nov 14 2019

			672 is in sequence because 672 / tau(672) - sigma(672) / 672 = 672 / 24 - 2016 / 672 = 25 (integer).

Cf. A000005, A000203, A216793, A245776, A245777, A245779, A245782.

[n: n in [1..100000] | (Denominator((n/(#[d: d in Divisors(n)])) - (SumOfDivisors(n)/n)) eq 1)]

select(n -> type(n/numtheory:-tau(n) - numtheory:-sigma(n)/n,integer), [$1..10^8]); # Robert Israel, Aug 03 2014

for(n=1,10^8,s=n/numdiv(n);t=sigma(n)/n;if(floor(s-t)==s-t,print1(n,", "))) \\ Derek Orr, Aug 01 2014

A245777(a(n)) = 1.

a(8)-a(13) from Giovanni Resta, Jul 13 2015

A245786 Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer.

Original entry on oeis.org

1, 672, 4680, 30240, 23569920, 45532800, 275890944, 14182439040, 153003540480, 403031236608, 518666803200

Offset: 1

Jaroslav Krizek, Aug 15 2014

nonn

Numbers n such that A245784(n) / A245785(n) =  (n / A000005(n) + A000203(n) / n) is an integer.
Sequence of numbers k(n): 2, 31, 101, 319, 73660, 118579, …
Conjecture: Subsequence of A216793.
Refactorable multiply-perfect numbers (A245782) are members of this sequence.
a(12) > 10^13. - Giovanni Resta, Jul 13 2015

			672 is in sequence because 672/tau(672) + sigma(672)/672 = 672/24 + 2016/672 = 31 (integer).

Cf. A000005, A000203, A245784, A245785, A245782.

[n: n in [1..100000] | (Denominator((n/(#[d: d in Divisors(n)])) + (SumOfDivisors(n)/n)) eq 1)]

for(n=1, 10^8, s=n/numdiv(n); t=sigma(n)/n; if(floor(s+t)==s+t, print1(n, ", "))) \\ Derek Orr, Aug 15 2014

A245785(a(n)) = 1.

a(7)-a(11) from Giovanni Resta, Jul 13 2015

A378267 Numbers k that have a record number of common divisors with sigma(k).

Original entry on oeis.org

1, 6, 24, 120, 672, 4320, 26208, 30240, 524160, 2178540, 8714160, 8910720, 17428320, 45532800, 132723360, 208565280, 240589440, 470564640, 668304000, 1307124000, 5228496000, 10805558400, 14182439040, 31998395520, 159991977600

Offset: 1

Amiram Eldar, Nov 21 2024

Indices of records in A073802.
This sequence is infinite since A073802 is unbounded. For example, for any odd number m we have A073802(2^(m-1)*(2^m-1)) >= A000005(m) and the number of divisors of odd numbers is unbounded.
The corresponding record values are 1, 4, 6, 16, 24, 40, 60, 96, 144, 216, 240, 336, ... .
a(26) <= 799959888000.

Cf. A000005, A000203 (sigma), A014567, A073802, A216793.

seq[kmax_] := Module[{d, dmax = 0, s = {}}, Do[d = DivisorSigma[0, GCD[k, DivisorSigma[1, k]]]; If[d > dmax, dmax = d; AppendTo[s, k]], {k, 1, kmax}]; s]; seq[10^6]

lista(kmax) = {my(d, dmax = 0); for(k = 1, kmax, d = numdiv(gcd(k, sigma(k))); if(d > dmax, dmax = d; print1(k, ", ")));}

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

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A245778 Numbers n such that k(n) = n/tau(n) - sigma(n)/n is an integer.

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A245786 Numbers n such that k(n) = (n/tau(n) + sigma(n)/n) is an integer.

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A378267 Numbers k that have a record number of common divisors with sigma(k).

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