A009242 -id:A009242 - 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

A064987 a(n) = n*sigma(n).

Original entry on oeis.org

1, 6, 12, 28, 30, 72, 56, 120, 117, 180, 132, 336, 182, 336, 360, 496, 306, 702, 380, 840, 672, 792, 552, 1440, 775, 1092, 1080, 1568, 870, 2160, 992, 2016, 1584, 1836, 1680, 3276, 1406, 2280, 2184, 3600, 1722, 4032, 1892, 3696, 3510, 3312, 2256, 5952

Offset: 1

Vladeta Jovovic, Oct 30 2001

Dirichlet convolution of sigma_2(n)=A001157(n) with phi(n)=A000010(n). - Vladeta Jovovic, Oct 27 2002
Equals row sums of triangle A143311 and of triangle A143308. - Gary W. Adamson, Aug 06 2008
a(n) is also the sum of all n's present in A244580, or in other words, a(n) is also the volume (or number of cubes) below the terraces of the n-th level of the staircase described in A244580 (see also A237593). - Omar E. Pol, Oct 11 2018
If n is a superperfect number then sigma(n) is a Mersenne prime and a(n) is a perfect number, a(A019279(k)) = A000396(k), k >= 1, assuming there are no odd perfect numbers. - Omar E. Pol, Apr 15 2020

B. C. Berndt, Ramanujan's theory of theta-functions, Theta functions: from the classical to the modern, Amer. Math. Soc., Providence, RI, 1993, pp. 1-63. MR 94m:11054. see page 43.
G. H. Hardy, Ramanujan: twelve lectures on subjects suggested by his life and work, AMS Chelsea Publishing, Providence, Rhode Island, 2002, pp. 166-167.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Joerg Arndt, On computing the generalized Lambert series, arXiv:1202.6525v3 [math.CA], (2012).
Passawan Noppakaew and Prapanpong Pongsriiam, Product of Some Polynomials and Arithmetic Functions, J. Int. Seq. (2023) Vol. 26, Art. 23.9.1.
Michel Planat, Twelve-dimensional Pauli group contextuality with eleven rays, arXiv:1201.5455 [quant-ph], 2012.

Main diagonal of A319073.

Cf. A000203, A038040, A002618, A000010, A001157, A143308, A143311, A004009, A006352, A000594, A126832, A069097 (Mobius transform), A001001 (inverse Mobius transform), A237593, A244580.

a:=List([1..50],n->n*Sigma(n));; Print(a); # Muniru A Asiru, Jan 01 2019

a064987 n = a000203 n * n  -- Reinhard Zumkeller, Jan 21 2014

[n*SumOfDivisors(n): n in [1..70]]; // Vincenzo Librandi, Jan 01 2019

with(numtheory): [n*sigma(n)$n=1..50]; # Muniru A Asiru, Jan 01 2019

# DivisorSigma[1,#]&/@Range[80]  (* Harvey P. Dale, Mar 12 2011 *)

numlib::sigma(n)*n$ n=1..81 // Zerinvary Lajos, May 13 2008

PARI
```
{a(n) = if ( n==0, 0, n * sigma(n))}
```

{ for (n=1, 1000, write("b064987.txt", n, " ", n*sigma(n)) ) } \\ Harry J. Smith, Oct 02 2009

Multiplicative with a(p^e) = p^e * (p^(e+1) - 1) / (p - 1).
G.f.: Sum_{n>0} n^2*x^n/(1-x^n)^2. - Vladeta Jovovic, Oct 27 2002
G.f.: phi_{2, 1}(x) where phi_{r, s}(x) = Sum_{n, m>0} m^r * n^s * x^{m*n}. - Michael Somos, Apr 02 2003
G.f. is also (Q - P^2) / 288 where P, Q are Ramanujan Lambert series. - Michael Somos, Apr 02 2003. See the Hardy reference, p. 136, eq. (10.5.4) (with a proof). For Q and P, (10.5.6) and (10.5.5), see E_4 A004009 and E_2 A006352, respectively. - Wolfdieter Lang, Jan 30 2017
Convolution of A000118 and A186690. Dirichlet convolution of A000027 and A000290. - Michael Somos, Mar 25 2012
Dirichlet g.f.: zeta(s-1)*zeta(s-2). - R. J. Mathar, Feb 16 2011
a(n) = A009194(n)*A009242(n). - Michel Marcus, Oct 23 2013
a(n) (mod 5) = A126832(n) = A000594(n) (mod 5). See A126832 for references. - Wolfdieter Lang, Feb 03 2017
L.g.f.: Sum_{k>=1} k*x^k/(1 - x^k) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 13 2017
Sum_{k>=1} 1/a(k) = 1.4383899259334187832765458631783591251241657856627653748389234270650138768... - Vaclav Kotesovec, Sep 20 2020
From Peter Bala, Jan 21 2021: (Start)
G.f.: Sum_{n >= 1} n*q^n*(1 + q^n)/(1 - q^n)^3 (use the expansion x*(1 + x)/(1 - x)^3 = x + 2^2*x^2 + 3^2*x^3 + 4^2*x^4 + ...).
A faster converging g.f.: Sum_{n >= 1} q^(n^2)*( n^3*q^(3*n) - (n^3 + 3*n^2 - n)*q^(2*n) - (n^3 - 3*n^2 - n)*q^n + n^3 )/(1 - q^n)^3 - differentiate equation 5 in Arndt w.r.t. both x and q and then set x = 1. (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} sigma_2(gcd(n,k)).
a(n) = Sum_{k=1..n} sigma_2(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)
From Peter Bala, Jan 22 2024: (Start)
a(n) = Sum_{1 <= j, k <= n} sigma_1( gcd(j, k, n) ).
a(n) = Sum_{d divides n} sigma_1(d)*J_2(n/d) = Sum_{d divides n} sigma_2(d)* phi(n/d), where the Jordan totient function J_2(n) = A007434(n). (End)

A173441 Number of divisors d of n such that sigma(d) divides n.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 2, 1, 3, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 4, 1, 1, 1, 5, 1, 1, 1, 1, 1, 2, 1, 1, 1, 1, 1, 4, 1, 1, 1, 1, 1, 2, 1, 1

Offset: 1

Jaroslav Krizek, Feb 18 2010

nonn

From Robert Israel, Oct 11 2017: (Start)
a(n) >= 1 since d=1 is always included.
a(n) = 1 if n is in A000961.
a(n) > 1 if n is in A097603.  The first n not in A097603 such that a(n) > 1 is 117. (End)

			For n = 12, a(12) = 4; divisors of 12: 1, 2, 3, 4, 6, 12; corresponding sigma(d):1, 3, 4, 7, 12, 28; sigma(d) divides n for 4 divisors d: 1, 2, 3, 6.

Robert Israel, Table of n, a(n) for n = 1..10000

Cf. A000005, A000961, A009242, A097603, A173442.

f:= proc(n) nops(select(t -> n mod numtheory:-sigma(t) = 0, numtheory:-divisors(n))) end proc:
map(f, [$1..100]); # Robert Israel, Oct 11 2017

a[n_] := Select[Divisors[n], Divisible[n, DivisorSigma[1, #]]&] // Length;
Array[a, 100] (* Jean-François Alcover, Jun 05 2020 *)

a(n) = sumdiv(n, d, !(n % sigma(d))); \\ Michel Marcus, Oct 11 2017

a(n) = A000005(n) - A173442(n). - A-number inserted by R. J. Mathar, Mar 06 2010

Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k) = Sum_{n>=1} 1/A009242(n) = 1.605582... . - Amiram Eldar, Mar 28 2024

A334783 a(n) = Sum_{d|n} lcm(d, sigma(d)).

Original entry on oeis.org

1, 7, 13, 35, 31, 31, 57, 155, 130, 127, 133, 143, 183, 231, 163, 651, 307, 382, 381, 575, 741, 535, 553, 383, 806, 735, 1210, 315, 871, 631, 993, 2667, 673, 1231, 1767, 3770, 1407, 1527, 2379, 1055, 1723, 1599, 1893, 1487, 1450, 2215, 2257, 2367, 2850, 5552

Offset: 1

Jaroslav Krizek, May 10 2020

			a(6) = lcm(1, sigma(1)) + lcm(2, sigma(2)) + lcm(3, sigma(3)) + lcm(6, sigma(6)) = lcm(1, 1) + lcm(2, 3) + lcm(3, 4) + lcm(6, 12) = 1 + 6 + 12 + 12 = 31.

Robert Israel, Table of n, a(n) for n = 1..10000
Robert Israel, Plot of a(n)/n^2 for n=1..20000

Cf. A334490 (Sum_{d|n} gcd(d, sigma(d))), A334782 (Sum_{d|n} lcm(d, tau(d))).

Cf. A000005 (tau(n)), A000203 (sigma(n)), A009242 (lcm(n, sigma(n))).

[&+[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

N:= 100: # for a(1)..a(N)
V:= Vector(N):
for d from 1 to N do
  t:= ilcm(d,numtheory:-sigma(d));
  R:= [seq(i,i=d..N,d)];
  V[R]:= V[R] +~ t;
od:
convert(V,list); # Robert Israel, May 13 2020

a[n_] := DivisorSum[n, LCM[#, DivisorSigma[1, #]] &]; Array[a, 100] (* Amiram Eldar, May 10 2020 *)

a(n) = sumdiv(n, d, lcm(d, sigma(d))); \\ Michel Marcus, May 10 2020

a(p) = p^2 + p + 1 for p = primes (A000040).

A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

Original entry on oeis.org

135, 546, 672, 9585, 24570, 51510, 63855, 190008, 251370, 323730, 372438, 486180, 510570, 723550, 819000, 1058910, 1282365, 1284192, 1356030, 3506390, 5416200, 5604480, 6298625, 15593760, 17813250, 18009000, 20740590, 26759370, 27027000, 27081000, 29795040

Offset: 1

Marius A. Burtea, Oct 23 2019

nonn

Composite numbers k verifying equation A009230(k) + A009262(k) = A009242(k).
For any prime number p >= 3 the equality lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)) is satisfied.
The sequence terms are the composite numbers for which the equality is true.

			For k = 135 = 3^3 * 5, tau(k) = 4 * 2 = 2^3, phi(k) = 2 * 3^2 * 4 = 2^3 * 3^2 , sigma(k) = 2^4 * 3 * 5, lcm(k, tau(k)) + lcm(k, phi(k)) =  2^3 * 3^3 * 5 + 2^3 * 3^3 * 5 = 2^4 * 3^3 * 5 and lcm(k, sigma(k)) = lcm(3^3 * 5, 2^4 * 3 * 5) = 2^4 * 3^3 * 5.

Cf. A000005, A000010, A000203, A009242, A009230, A009262, A326416.

[k: k in [1..6000000]| not IsPrime(k) and Lcm(k,NumberOfDivisors(k))+Lcm(k,EulerPhi(k)) eq Lcm(k,SumOfDivisors(k))];

aQ[n_] := CompositeQ[n] && LCM[n, EulerPhi[n]] + LCM[n, DivisorSigma[0, n]] == LCM[n, DivisorSigma[1, n]]; Select[Range[3*10^6], aQ] (* Amiram Eldar, Oct 23 2019 *)

isok(k) = !isprime(k) && (lcm(k, numdiv(k)) + lcm(k, eulerphi(k)) == lcm(k, sigma(k))); \\ Michel Marcus, Oct 24 2019

A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) 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, 18, 468, 9360, 10880, 79360, 84480, 387072, 777216, 3801600, 7282688, 15037440, 17418240, 27067392, 65544192, 752903424, 1218032640, 4227842304, 4737761280, 6410638080, 11949932544, 19327057920, 26372530800, 37645171200, 224956569600, 243520929792, 876611248128

Offset: 1

Jaroslav Krizek, Mar 03 2019

nonn

Numbers n such that A009242(n) = A000005(n) * A000203(n) = A064840(n).
Also numbers n such that A017666(n) = denominator(sigma(n)/n) = tau(n) = A000005(n).
a(29) > 10^12. - Giovanni Resta, Mar 04 2019

			18 is a term because lcm(sigma(18), 18) = lcm(39, 18) = 234 = tau(18) * sigma(18) = 6 * 39.

Cf. A000005, A000203, A009242, A064840.

Cf. A069810 (gcd(sigma(n), n) = tau(n)).

[n: n in [1..1000000] | LCM(SumOfDivisors(n), n) eq NumberOfDivisors(n)* SumOfDivisors(n)]

Select[Range[1000000], LCM[DivisorSigma[1, #], #] == DivisorSigma[0, #] * DivisorSigma[1, #]&] (* Vaclav Kotesovec, Mar 04 2019 *)

isok(n) = my(sn = sigma(n)); lcm(sn, n) == sn*numdiv(n); \\ Michel Marcus, Mar 04 2019

a(13)-a(16) from Vaclav Kotesovec, Mar 04 2019
a(17) from Daniel Suteu, Mar 04 2019
a(18)-a(28) from Giovanni Resta, Mar 04 2019

A334805 a(n) = Product_{d|n} lcm(d, sigma(d)) where sigma(k) is the sum of divisors of k (A000203).

Original entry on oeis.org

1, 6, 12, 168, 30, 864, 56, 20160, 1404, 16200, 132, 2032128, 182, 56448, 43200, 9999360, 306, 23654592, 380, 190512000, 451584, 313632, 552, 29262643200, 23250, 596232, 1516320, 88510464, 870, 100776960000, 992, 20158709760, 836352, 1685448, 2822400

Offset: 1

Jaroslav Krizek, Jun 26 2020

nonn

			a(6) = lcm(1, sigma(1)) * lcm(2, sigma(2)) * lcm(3, sigma(3)) * lcm(6, sigma(6)) = lcm(1, 1) * lcm(2, 3) * lcm(3, 4) * lcm(6, 12) = 1 * 6 * 12 * 12 = 864.

Cf. A334783 (Sum_{d|n} lcm(d, sigma(d))), A334491 (Product_{d|n} gcd(d, sigma(d))).

Cf. A000203 (sigma(n)), A009242 (lcm(n, sigma(n))), A036690.

[&*[LCM(d, &+Divisors(d)): d in Divisors(n)]: n in [1..100]]

a[n_] := Product[LCM[d, DivisorSigma[1, d]], {d, Divisors[n]}]; Array[a, 35] (* Amiram Eldar, Jun 27 2020 *)

a(n) = my(d=divisors(n)); prod(k=1, #d, lcm(d[k], sigma(d[k]))); \\ Michel Marcus, Jun 27 2020

a(p) = p^2 + p for p = primes (A000040).

cp's OEIS Frontend

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

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A064987 a(n) = n*sigma(n).

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A173441 Number of divisors d of n such that sigma(d) divides n.

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A334783 a(n) = Sum_{d|n} lcm(d, sigma(d)).

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A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

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A306655 Numbers n such that lcm(sigma(n), n) = tau(n) * sigma(n) where sigma(k) = the sum of the divisors of k (A000203) and tau(k) = the number of divisors of k (A000005).

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