A064840 -id:A064840 - cp's OEIS frontend

A062354 a(n) = sigma(n)*phi(n).

1, 3, 8, 14, 24, 24, 48, 60, 78, 72, 120, 112, 168, 144, 192, 248, 288, 234, 360, 336, 384, 360, 528, 480, 620, 504, 720, 672, 840, 576, 960, 1008, 960, 864, 1152, 1092, 1368, 1080, 1344, 1440, 1680, 1152, 1848, 1680, 1872, 1584, 2208, 1984, 2394, 1860

Offset: 1

Jason Earls, Jul 06 2001

Let G_n be the group of invertible 2 X 2 matrices mod n (sequence A000252). a(n) is the number of conjugacy classes in G_n. - Dan Fux (dan.fux(AT)OpenGaia.com or danfux(AT)OpenGaia.com), Nov 13 2001
a(n) = Sum_{d|n} phi(n*d). - Vladeta Jovovic, Apr 17 2002
Apparently the Mobius transform of A062952. - R. J. Mathar, Oct 01 2011

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.

T. D. Noe, Table of n, a(n) for n=1..10000
Vaclav Kotesovec, Plot of Sum_{k=1..n} a(k) / (Pi^2 * n^3 / 18) for n = 1..1000000
J.-L. Nicolas and Jonathan Sondow, Ramanujan, Robin, highly composite numbers, and the Riemann Hypothesis, arXiv:1211.6944 [math.HO], 2012, to appear in RAMA125 Proceedings, Contemp. Math.

Cf. A000010, A000203, A000252, A062355, A064840, A093827.

Table[EulerPhi[n] DivisorSigma[1, n], {n, 1, 80}] (* Carl Najafi, Aug 16 2011 *)

a(n)=sigma(n)*eulerphi(n); vector(150,n,a(n))

Multiplicative with a(p^e) = p^(e-1)*(p^(e+1)-1). - Vladeta Jovovic, Apr 17 2002
Dirichlet g.f.: zeta(s-1)*zeta(s-2)*product_{primes p} (1-p^(1-s)-p^(-s)+p^(2-2s)). - R. J. Mathar, Oct 01 2011, corrected by Vaclav Kotesovec, Dec 17 2019
6/Pi^2 < a(n)/n^2 < 1 for n > 1. - Jonathan Sondow, Mar 06 2014
Sum_{k=1..n} a(k) ~ c * Pi^2 * n^3 / 18, where c = A330523 = Product_{primes p} (1 - 1/p^2 - 1/p^3 + 1/p^4) = 0.535896... - Vaclav Kotesovec, Dec 17 2019
Sum_{n>=1} 1/a(n) = 1.7865764... (A093827). - Amiram Eldar, Aug 20 2020
a(n)/n^2 > 8/Pi^2 for odd n. - M. F. Hasler, Jul 08 2025

A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

Original entry on oeis.org

1, 2, 4, 6, 8, 8, 12, 16, 18, 16, 20, 24, 24, 24, 32, 40, 32, 36, 36, 48, 48, 40, 44, 64, 60, 48, 72, 72, 56, 64, 60, 96, 80, 64, 96, 108, 72, 72, 96, 128, 80, 96, 84, 120, 144, 88, 92, 160, 126, 120, 128, 144, 104, 144, 160, 192, 144, 112, 116, 192, 120, 120, 216, 224

Offset: 1

Jason Earls, Jul 06 2001

a(n) = sum of gcd(k-1,n) for 1 <= k <= n and gcd(k,n)=1 (Menon's identity).
For n = 2^(4*k^2 - 1), k >= 1, the terms of the sequence are square and for n = 2^((3*k + 2)^3 - 1), k >= 1, the terms of the sequence are cubes. - Marius A. Burtea, Nov 14 2019
Sum_{k>=1} 1/a(k) diverges. - Vaclav Kotesovec, Sep 20 2020

D. M. Burton, Elementary Number Theory, Allyn and Bacon Inc., Boston MA, 1976, Prob. 7.2 12, p. 141.
P. K. Menon, On the sum gcd(a-1,n) [(a,n)=1], J. Indian Math. Soc. (N.S.), 29 (1965), 155-163.
József Sándor, On Dedekind's arithmetical function, Seminarul de teoria structurilor (in Romanian), No. 51, Univ. Timișoara, 1988, pp. 1-15. See p. 11.
József Sándor, Some diophantine equations for particular arithmetic functions (in Romanian), Seminarul de teoria structurilor, No. 53, Univ. Timișoara, 1989, pp. 1-10. See p. 8.

Antti Karttunen, Table of n, a(n) for n = 1..65537 (terms 1..1000 from Harry J. Smith)
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Vaclav Kotesovec, Graph - the asymptotic ratio (250000000 terms)
R. J. Mathar, Survey of Dirichlet series of multiplicative arithmetic functions, arXiv:1106.4038 [math.NT], 2011-2012, Section 3.15.
R. Sivaramakrishnan, Problem E 1962, Elementary Problems, The American Mathematical Monthly, Vol. 74, No. 2 (1967), p. 198; Solution, ibid., Vol. 75, No. 5 (1968), p. 550.
Marius Tarnauceanu, A generalization of the Menon's identity, arXiv:1109.2198 [math.GR], 2011-2012.
Laszlo Toth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.
Wikipedia, Arithmetic function (Menon's identity).

Cf. A003557, A173557, A061468, A062816, A079535, A062949 (inverse Mobius transform), A304408, A318519, A327169 (number of times n occurs in this sequence).

Cf. A062354, A064840.

[NumberOfDivisors(n)*EulerPhi(n):n in [1..65]]; // Marius A. Burtea, Nov 14 2019

seq(tau(n)*phi(n), n=1..64); # Zerinvary Lajos, Jan 22 2007

Table[EulerPhi[n] DivisorSigma[0, n], {n, 80}] (* Carl Najafi, Aug 16 2011 *)
f[p_, e_] := (e+1)*(p-1)*p^(e-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 21 2020 *)

a(n)=numdiv(n)*eulerphi(n); vector(150,n,a(n))

{ for (n=1, 1000, write("b062355.txt", n, " ", numdiv(n)*eulerphi(n)) ) } \\ Harry J. Smith, Aug 05 2009

for(n=1, 100, print1(direuler(p=2, n, (1 - 2*X + p*X^2)/(1 - p*X)^2)[n], ", ")) \\ Vaclav Kotesovec, Jun 15 2020

Dirichlet convolution of A047994 and A000010. - R. J. Mathar, Apr 15 2011
a(n) = A000005(n)*A000010(n). Multiplicative with a(p^e) = (e+1)*(p-1)*p^(e-1). - R. J. Mathar, Jun 23 2018
a(n) = A173557(n) * A318519(n) = A003557(n) * A304408(n). - Antti Karttunen, Sep 16 2018 & Sep 20 2019
From Vaclav Kotesovec, Jun 15 2020: (Start)
Let f(s) = Product_{primes p} (1 - 2*p^(-s) + p^(1-2*s)).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ n^2 * (f(2)*(log(n)/2 + gamma - 1/4) + f'(2)/2), where f(2) = A065464 = Product_{primes p} (1 - 2/p^2 + 1/p^3) = 0.42824950567709444...,
f'(2) = 2 * A065464 * A335707 = f(2) * Sum_{primes p} 2*log(p) / (p^2 + p - 1) = 0.35866545223424232469545420783620795... and gamma is the Euler-Mascheroni constant A001620. (End)
From Amiram Eldar, Mar 02 2021: (Start)
a(n) >= n (Sivaramakrishnan, 1967).
a(n) >= sigma(n), for odd n (Sándor, 1988).
a(n) >= phi(n) + n - 1 (Sándor, 1989) (End)
From Richard L. Ollerton, May 07 2021: (Start)
a(n) = Sum_{k=1..n} uphi(gcd(n,k)), where uphi(n) = A047994(n).
a(n) = Sum_{k=1..n} uphi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). (End)

A009205 a(n) = gcd(d(n), sigma(n)).

Original entry on oeis.org

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

Offset: 1

David W. Wilson

nonn

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

Cf. A000005, A000203, A009191, A009194, A009278, A064840, A087801, A286360.

Table[GCD[DivisorSigma[0,n],DivisorSigma[1,n]],{n,120}] (* Harvey P. Dale, Dec 05 2017 *)

A009205(n) = gcd(numdiv(n),sigma(n)); \\ Antti Karttunen, May 22 2017

from math import prod, gcd
from sympy import factorint
def A009205(n):
    f = factorint(n).items()
    return gcd(prod(e+1 for p, e in f),prod((p**(e+1)-1)//(p-1) for p,e in f)) # Chai Wah Wu, Jul 27 2023

a(n) = A064840(n)/A009278(n). - Amiram Eldar, Jan 31 2025

Data section extended to 120 terms by Antti Karttunen, May 22 2017

A064945 a(n) = Sum_{i|n, j|n, j >= i} i.

Original entry on oeis.org

1, 4, 5, 11, 7, 22, 9, 26, 18, 30, 13, 64, 15, 38, 38, 57, 19, 82, 21, 87, 48, 54, 25, 156, 38, 62, 58, 109, 31, 179, 33, 120, 68, 78, 68, 244, 39, 86, 78, 213, 43, 224, 45, 153, 143, 102, 49, 348, 66, 166, 98, 175, 55, 268, 96, 267, 108, 126, 61, 542, 63, 134, 181

Offset: 1

Vladeta Jovovic, Oct 28 2001

			a(6) = dot_product(4,3,2,1)*(1,2,3,6) = 4*1+3*2+2*3+1*6 = 22.

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A060640, A064944, A064946, A064947, A064948, A064949.
Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.
Cf. A386911.

a064945 = sum . zipWith (*) [1..] . reverse . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015

with(numtheory): seq(add((tau(n)-i+1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064945[n_] := #.Range[Length[#], 1, -1] & [Divisors[n]];
Array[A064945, 100] (* Paolo Xausa, Aug 07 2025 *)

a(n) = my(d=divisors(n), t=length(d)); sum(i=1, t, (t - i + 1)*d[i]); \\ Harry J. Smith, Oct 01 2009

a(n)=my(d=divisors(n)); sum(i=1,#d,(#d+1-i)*d[i]) \\ Charles R Greathouse IV, Jun 10 2015

from sympy import divisors, divisor_sigma
def A064945(n): return (divisor_sigma(n,0)+1)*divisor_sigma(n)-sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

a(n) = Sum_{i=1..tau(n)} (tau(n)-i+1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} (A000005(n)-i+1)*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 02 2025: (Start)
a(n) = Sum_{d|n} d*A135539(n,d).
a(n) = A064947(n) + A000203(n).
a(n) = (A064949(n) + A000203(n))/2.
a(n) = A064949(n) - A064947(n).
a(n) = A337360(n) - A064944(n).
a(n) = A064840(n) - A064946(n). (End)

A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

Original entry on oeis.org

1, 5, 7, 17, 11, 38, 15, 49, 34, 60, 23, 132, 27, 82, 82, 129, 35, 191, 39, 207, 112, 126, 47, 384, 86, 148, 142, 283, 59, 469, 63, 321, 172, 192, 172, 666, 75, 214, 202, 597, 83, 640, 87, 435, 403, 258, 95, 1016, 162, 485, 262, 511, 107, 812, 264, 813, 292, 324

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = max(1,1)+max(1,2)+max(1,3)+max(1,6)+max(2,2)+max(2,3)+max(2,6)+max(3,3)+max(3,6)+max(6,6)=38, or a(6) = dot_product(1,2,3,4)*(1,2,3,6)=1*1+2*2+3*3+4*6=38.

Paolo Xausa, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Harry J. Smith)

Cf. A060640, A064945, A064946, A064947, A064948, A064949.

Cf. A027750, A135539, A000005, A000203, A064840, A337360, A337297.

a064944 = sum . zipWith (*) [1..] . a027750_row'
-- Reinhard Zumkeller, Jul 14 2015

with(numtheory): seq(add(i*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064944[n_] := #.Range[Length[#]] & [Divisors[n]];
Array[A064944, 100] (* Paolo Xausa, Aug 07 2025 *)

a(n) = my(d=divisors(n)); sum(i=1, length(d), i*d[i]); \\ Harry J. Smith, Sep 30 2009

from sympy import divisors
def A064944(n): return sum(a*b for a, b in enumerate(divisors(n),1)) # Chai Wah Wu, Aug 07 2025

a(n) = Sum_{i=1..tau(n)} i*d_i, where {d_i}, i=1..tau(n) is the increasing sequence of divisors of n.
a(n) = Sum_{i=1..A000005(n)} i*A027750(n, i). - Michel Marcus, Jun 10 2015
From Ridouane Oudra, Aug 01 2025: (Start)
a(n) = Sum_{d|n} (n/d)*A135539(n,d).
a(n) = A064946(n) + A000203(n).
a(n) = (A064948(n) + A000203(n))/2.
a(n) = A337360(n) - A064945(n).
a(n) = A064948(n) - A064946(n).
a(n) = A064840(n) - A064947(n). (End)

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

A064949 a(n) = Sum_{i|n, j|n} min(i,j).

Original entry on oeis.org

1, 5, 6, 15, 8, 32, 10, 37, 23, 42, 14, 100, 16, 52, 52, 83, 20, 125, 22, 132, 64, 72, 26, 252, 45, 82, 76, 162, 32, 286, 34, 177, 88, 102, 88, 397, 40, 112, 100, 336, 44, 352, 46, 222, 208, 132, 50, 572, 75, 239, 124, 252, 56, 416, 120, 414, 136, 162, 62, 916, 64

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = dot_product(7,5,3,1)*(1,2,3,6) = 7*1 + 5*2 + 3*3 + 1*6 = 32.

Antti Karttunen, Table of n, a(n) for n = 1..20000 (first 1000 terms from Harry J. Smith)

Cf. A000005, A000203, A064944, A064945, A135539, A064840.

with(numtheory): seq(add((2*tau(n)-2*i+1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

Array[Function[{t, d}, Total@ MapIndexed[#1 (2 t - 2 First[#2] + 1) &, d]] @@ {DivisorSigma[0, #], Divisors[#]} &, 61] (* Michael De Vlieger, Oct 25 2021 *)

a(n) = { my(d=divisors(n), t=length(d)); sum(i=1, t, (2*t - 2*i + 1)*d[i]) } \\ Harry J. Smith, Oct 01 2009

A064949(n) = { my(i=0, u=numdiv(n)); sumdiv(n,d,i++; (((2*u)-(2*i))+1)*d); }; \\ Antti Karttunen, Nov 14 2021

a(n) = Sum_{i=1..tau(n)} (2*tau(n)-2*i+1)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.
a(n) = Sum_{i=1..n} A135539(n,i)^2. - Ridouane Oudra, Oct 25 2021
a(n) = A000203(n) * (2*A000005(n)+1) - 2*A064944(n). - Amiram Eldar, Jan 13 2025
From Ridouane Oudra, Aug 13 2025: (Start)
a(n) = A064945(n) + A064947(n).
a(n) = 2*A064947(n) + A000203(n).
a(n) = 2*A064945(n) - A000203(n).
a(n) = 2*A064840(n) - A064948(n). (End)

A304412 If n = Product (p_j^k_j) then a(n) = Product ((p_j + 1)*(k_j + 1)).

Original entry on oeis.org

1, 6, 8, 9, 12, 48, 16, 12, 12, 72, 24, 72, 28, 96, 96, 15, 36, 72, 40, 108, 128, 144, 48, 96, 18, 168, 16, 144, 60, 576, 64, 18, 192, 216, 192, 108, 76, 240, 224, 144, 84, 768, 88, 216, 144, 288, 96, 120, 24, 108, 288, 252, 108, 96, 288, 192, 320, 360, 120, 864, 124, 384, 192, 21, 336, 1152, 136, 324

Offset: 1

Ilya Gutkovskiy, May 12 2018

			a(36) = a(2^2*3^2) = (2 + 1)*(2 + 1) * (3 + 1)*(2 + 1) = 108.

Andrew Howroyd, Table of n, a(n) for n = 1..1000
Ilya Gutkovskiy, Scatter plot of a(n) up to n=50000.
Vaclav Kotesovec, Graph - the asymptotic ratio (10000 terms)
Index entries for sequences computed from exponents in factorization of n.
Index entries for sequences computed from indices in prime factorization.

Cf. A000005, A000026, A000040, A000203, A000302 (numbers n such that a(n) is odd), A001221, A001615, A003959, A005117, A007947, A008864, A045967, A048250, A064549, A064840, A304407, A304408, A304409, A304411.

Cf. A065463.

a[n_] := Times @@ ((#[[1]] + 1) (#[[2]] + 1) & /@ FactorInteger[n]); a[1] = 1; Table[a[n], {n, 68}]
Table[DivisorSigma[0, n] Total[Select[Divisors[n], SquareFreeQ]], {n, 68}]

a(n)={numdiv(n)*sumdiv(n, d, moebius(d)^2*d)} \\ Andrew Howroyd, Jul 24 2018

a(n) = A000005(n)*A048250(n) = A000005(n)*A000203(A007947(n)).
a(p^k) = (p + 1)*(k + 1) where p is a prime and k > 0.
a(n) = 2^omega(n)*Product_{p|n} (p + 1) if n is a squarefree (A005117), where omega() = A001221.
Dirichlet g.f.: zeta(s)^2 * Product_{p prime} (1 + 2/p^(s-1) - 1/p^(2*s-1)). - Amiram Eldar, Sep 17 2023
From Vaclav Kotesovec, May 06 2025: (Start)
Let f(s) = Product_{p prime} (p^s-p)^2 * (p^(2*s)+2*p^(s+1)-p) / (p^(2*s) * (p^s-1)^2).
Dirichlet g.f.: zeta(s-1)^2 * f(s).
Sum_{k=1..n} a(k) ~ ((2*log(n) + 4*gamma - 1)*f(2) + 2*f'(2)) * n^2/4, where
f(2) = A065463 = Product_{p prime} (1 - 1/(p*(p+1))) = 0.704442200999165592736603350326637210188586431417098049414226842591...,
f'(2) = f(2) * Sum_{p prime} 2*(2*p^2-1)*log(p) / ((p^2-1)*(p^2+p-1)) = f(2) * 1.799151495460164053607059266860868724519705035904425832307664926571...
and gamma is the Euler-Mascheroni constant A001620. (End)

A064946 a(n) = Sum_{i|n, j|n, j>i} j.

Original entry on oeis.org

0, 2, 3, 10, 5, 26, 7, 34, 21, 42, 11, 104, 13, 58, 58, 98, 17, 152, 19, 165, 80, 90, 23, 324, 55, 106, 102, 227, 29, 397, 31, 258, 124, 138, 124, 575, 37, 154, 146, 507, 41, 544, 43, 351, 325, 186, 47, 892, 105, 392, 190, 413, 53, 692, 192, 693, 212, 234, 59

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

			a(6) = dot_product(0,1,2,3)*(1,2,3,6) = 0*1 + 1*2 + 2*3 + 3*6 = 26.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A064944, A064945, A064947, A064948, A064949.

Cf. A000005, A000203, A064840, A337297, A027750.

with(numtheory): seq(add((i-1)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

A064946[n_] := #.Range[Length[#]] & [Rest[Divisors[n]]];
Array[A064946, 100] (* Paolo Xausa, Aug 14 2025 *)

a(n) = my(d=divisors(n)); sum(i=2, length(d), (i - 1)*d[i]); \\ Harry J. Smith, Oct 01 2009

a(n) = Sum_{i=1..tau(n)} (i-1)*d_i, where {d_i}, i=1..tau(n), is the increasing sequence of the divisors of n.
a(n) = A064944(n) - A000203(n). - Amiram Eldar, Dec 23 2024
From Ridouane Oudra, Aug 06 2025: (Start)
a(n) = A064948(n) - A064944(n).
a(n) = A064840(n) - A064945(n).
a(n) = A337297(n) - A064947(n).
a(n) = (A064948(n) - A000203(n))/2. (End)

A064947 a(n) = Sum_{i|n, j|n, j>i} i.

Original entry on oeis.org

0, 1, 1, 4, 1, 10, 1, 11, 5, 12, 1, 36, 1, 14, 14, 26, 1, 43, 1, 45, 16, 18, 1, 96, 7, 20, 18, 53, 1, 107, 1, 57, 20, 24, 20, 153, 1, 26, 22, 123, 1, 128, 1, 69, 65, 30, 1, 224, 9, 73, 26, 77, 1, 148, 24, 147, 28, 36, 1, 374, 1, 38, 77, 120, 26, 168, 1, 93, 32, 165, 1, 411, 1, 44

Offset: 1

Vladeta Jovovic, Oct 28 2001

nonn

For given n, iterate a(n); a(a(n)); a(a(a(n))); ... Does this iterative process always lead to a(a(...(a(n))...)) = 1? - Ctibor O. Zizka, Apr 17 2008

No. For example, a(4) = 4, a(14) = 14, and a(99) = 99. - Jason Yuen, Jan 07 2025

			a(6) = dot_product(3,2,1,0)*(1,2,3,6) = 3*1 + 2*2 + 1*3 + 0*6 = 10.

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000005, A000203, A064840, A337297, A027750.

with(numtheory): seq(add((tau(n)-i)*sort(convert(divisors(n),'list'))[i],i=1..tau(n)), n=1..200);

Table[t = DivisorSigma[0, n]; Total@ MapIndexed[(t - First[#2])*#1 &, Divisors[n]], {n, 120}] (* Michael De Vlieger, Jan 07 2025 *)

a(n) = my(d=divisors(n), t=length(d)); sum(i=1, t - 1, (t - i)*d[i]); \\ Harry J. Smith, Oct 01 2009

a(n) = Sum_{i=1..tau(n)} (tau(n)-i)*d_i, where {d_i}, i=1..tau(n), is increasing sequence of divisors of n.
From Ridouane Oudra, Aug 07 2025: (Start)
a(n) = A064945(n) - A000203(n).
a(n) = A064840(n) - A064944(n).
a(n) = A064949(n) - A064945(n).
a(n) = A337297(n) - A064946(n).
a(n) = (A064949(n) - A000203(n))/2. (End)

cp's OEIS Frontend

A062354 a(n) = sigma(n)*phi(n).

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A062355 a(n) = d(n) * phi(n), where d(n) is the number of divisors function.

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A009205 a(n) = gcd(d(n), sigma(n)).

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A064945 a(n) = Sum_{i|n, j|n, j >= i} i.

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A064944 a(n) = Sum_{i|n, j|n, j >= i} j.

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

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