A018804 -id:A018804 - cp's OEIS frontend

A124647 a(n) = (2n + 1)*3^n.

1, 9, 45, 189, 729, 2673, 9477, 32805, 111537, 373977, 1240029, 4074381, 13286025, 43046721, 138706101, 444816117, 1420541793, 4519905705, 14334558093, 45328197213, 142958160441, 449795187729, 1412147682405, 4424729404869, 13839047287569, 43211719081593, 134718888901437

Offset: 0

Gary W. Adamson, Dec 22 2006

1 - 1/9 + 1/45 - 1/189 + ... = Pi/(2*sqrt(3)) = A093766. [Jolley eq 271].
If X_1,X_2,...,X_n are 3-blocks of a (4n+1)-set X then, for n>=1, a(n) is the number of (n+1)-subsets of X intersecting each X_i, (i=1,2,...,n). - Milan Janjic, Nov 23 2007
Sum_{k>=0} 1/a(k) = log(2+sqrt(3))*sqrt(3)/2 =  1.1405189944... - Jaume Oliver Lafont, Nov 30 2009

			a(3) = 189 = 7*(3^3).

L. B. W. Jolley, "Summation of Series", Dover Publications, 1961, p. 50

G. C. Greubel, Table of n, a(n) for n = 0..1000
Milan Janjic, Two Enumerative Functions
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023. See p. 14.
Index entries for linear recurrences with constant coefficients, signature (6,-9).

Cf. A000010, A000244, A018804, A036290, A081038, A093766.

[ (2*n+1)*3^n: n in [0..23] ]; // Klaus Brockhaus, Sep 23 2009

Table[3^n*(2*n+1), {n,0,30}] (* G. C. Greubel, May 01 2021 *)

[3^n*(2*n+1) for n in (0..30)] # G. C. Greubel, May 01 2021

G.f.: (1+3*x)/(1-3*x)^2. - Jaume Oliver Lafont, Mar 07 2009
a(n) = 6*a(n-1) - 9*a(n-2) for n > 1; a(0) = 1, a(1) = 9. - Klaus Brockhaus, Sep 23 2009
a(n) = 9*A081038(n-1) for n > 0. - Klaus Brockhaus, Sep 23 2009
a(n) = Sum_{i=1..2*3^n-1} gcd(i,2*3^n) = A018804(2*3^n) -2*3^n. This is an application of the multiplicative property of the gcd sum-function A018804. So we get: 2*3^0 * phi(3^n) + ... + 2*3^(n-1) * phi(3^1) + 2*3^n * phi(3^0)+3^0 * phi(2*3^n) + ... + 3^n * phi(2*3^0) - gcd(2*3^n,2*3^n) = a(n), where phi=A000010 is Euler's totient. A general formula is Sum_{i=1..2*p^n-1} gcd(i,2*p^n) = n*3*p^n * n - 3*n*p^(n-1) + p^n, for p an odd prime. This sequence correspondes to p=3. - Jeffrey R. Goodwin, Nov 10 2011
E.g.f.: exp(3*x)*(1 + 6*x). - Stefano Spezia, May 07 2023

More terms from Klaus Brockhaus, Sep 23 2009

A067911 Product of gcd(k,n) for 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 8, 5, 72, 7, 128, 81, 800, 11, 41472, 13, 6272, 30375, 32768, 17, 3359232, 19, 20480000, 750141, 247808, 23, 13759414272, 15625, 1384448, 1594323, 5035261952, 29, 30233088000000, 31, 2147483648, 235782657, 37879808

Offset: 1

Sharon Sela (sharonsela(AT)hotmail.com), Mar 10 2002

nonn

T. D. Noe, Table of n, a(n) for n = 1..500
Johann Cigler, Some remarks on the power product expansion of the q-exponential series, arXiv:2006.06242 [math.CO], 2020.
Gottfried Helms, A dream of a (number-) sequence, 2007-2009.
L. Toth, A survey of gcd-sum functions, J. Int. Seq. 13 (2010) # 10.8.1.

Cf. A051190, A051696.
In A018804 the product is replaced by sum.
Product of terms in n-th row of A050873.
Cf. A000010 (comments on product formulas).

with(numtheory): a := n -> mul(d^phi(n/d), d = divisors(n)):
seq(a(i), i = 1..34); # Peter Luschny, Apr 07 2013

a[n_] := Product[d^EulerPhi[n/d], {d, Divisors[n]}];
Array[a, 34] (* Jean-François Alcover, Jun 03 2019 *)

a(n) = prod(k=1, n, gcd(k, n)); \\ Michel Marcus, Aug 23 2016

A067911 = lambda n: mul(gcd(n,i) for i in range(n))
[A067911(n) for n in (1..34)] # Peter Luschny, Apr 07 2013

a(n) = Product_{d|n} d^phi(n/d). - Vladeta Jovovic, Mar 08 2004
a(n) = n*A051190(n). - Peter Luschny, Apr 07 2013
a(n) = Product_{k=1..n} (n/gcd(n,k))^(phi(gcd(n,k))/phi(n/gcd(n,k))) where phi = A000010. - Richard L. Ollerton, Nov 07 2021

Extended and edited by John W. Layman, Mar 14 2002

A078430 Sum of gcd(k^2,n) for 1 <= k <= n.

Original entry on oeis.org

1, 3, 5, 10, 9, 15, 13, 28, 33, 27, 21, 50, 25, 39, 45, 88, 33, 99, 37, 90, 65, 63, 45, 140, 145, 75, 153, 130, 57, 135, 61, 240, 105, 99, 117, 330, 73, 111, 125, 252, 81, 195, 85, 210, 297, 135, 93, 440, 385, 435, 165, 250, 105, 459, 189, 364, 185, 171, 117, 450, 121

Offset: 1

Vladeta Jovovic, Dec 30 2002

a(n) is the number of non-congruent solutions to x^2*y = 0 mod n. - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 17 2003

Row sums of triangle A245717. - Reinhard Zumkeller, Jul 30 2014

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Reinhard Zumkeller)
E. Krätzel, W. G. Nowak, and L. Tóth, On certain arithmetic functions involving the greatest common divisor, Cent. Eur. J. Math., 10 (2012), 761-774.
M. Kühleitner and W. G. Nowak, On a question of A. Schinzel: Omega estimates for a special type of arithmetic functions , arXiv: 1204.1146 [math.NT], 2012.
László Tóth, Menon's identity and arithmetical sums representing functions of several variables, Rend. Sem. Mat. Univ. Politec. Torino, 69 (2011), 97-110.

Cf. A018804, A069097, A069193.

Cf. A245717.

a078430 = sum . a245717_row  -- Reinhard Zumkeller, Jul 30 2014

Table[Sum[GCD[k^2,n],{k,n}],{n,70}] (* Harvey P. Dale, Sep 29 2014 *)
f[p_, e_] := If[EvenQ[e], p^(3*e/2) + p^(3*e/2 - 1), 2*p^((3*e - 1)/2)] - p^(e - 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Apr 28 2023 *)

a(n) = sum(k=1,n, gcd(k^2, n)); \\ Michel Marcus, Aug 03 2016

a(n) is multiplicative. G.f. for a(p^n), p a prime, is given by (1+(p-1)*x-p^2*x^2)/(1-p*x)/(1-p^3*x^2).
a(n) = n*Sum_{d|n} phi(d)*N(d)/d, where phi is Euler's totient function A000010 and N(n) is sequence A000188. - Laszlo Toth, Apr 15 2012
Multiplicative with a(p^e) = p^(3*e/2) + p^(3*e/2-1) - p^(e-1) if e is even, and 2*p^((3*e-1)/2) - p^(e-1) if e is odd. - Amiram Eldar, Apr 28 2023

A080997 The positive integers arranged in nonincreasing order of centrality (the fraction of n represented by the average gcd of n and the other positive integers).

Original entry on oeis.org

1, 2, 3, 4, 6, 5, 8, 12, 10, 7, 9, 15, 14, 18, 16, 20, 24, 11, 30, 13, 21, 28, 22, 36, 17, 40, 27, 26, 42, 32, 48, 25, 19, 60, 33, 35, 45, 44, 34, 23, 54, 56, 39, 72, 50, 38, 52, 84, 66, 70, 90, 63, 29, 80, 46, 31, 51, 64, 120, 55, 78, 96, 75, 68, 57, 108, 49, 88, 37, 65, 105

Offset: 1

Matthew Vandermast, Feb 28 2003

Equivalent descriptions of the centrality of n: 1) Probability that a randomly chosen product in the multiplication table for positive integers (A003991; see also A061017) is a multiple of n.
2) Probability taken over all exponential numerical bases that if the last digit of a number represents n, the number is a multiple of n. (For example, in base 10, the probability of a number that ends in 5 being a multiple of 5 is 1. Over all possible bases, the fraction of numbers ending in 5 that are multiples of 5 is the centrality of 5, 9/25 or .36.)
An infinite number of integers have the same centrality as at least one other integer. The only such examples in the first 114 terms of the sequence are 64 and 120, which share a centrality of .0625; they are listed in numerical order.

			The number 6 has a gcd of 1 with all numbers congruent to 1 or 5 modulo 6, 2 with all numbers congruent to 2 or 4 mod 6, 3 with all 3 mod 6 numbers and 6 with all numbers congruent to 0 mod 6. Its average gcd with other integers is 2.5 (A018804(6)/6), which represents 5/12 or .41666... of 6. This places 6 fifth in centrality among the integers, behind 1 (whose centrality is 1), 2 (.75), 3 (5/9 or .555...) and 4 (.5); it is therefore listed fifth in the sequence.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A018804, A080999 for a formula for the numerator of the unreduced centrality fraction. Other related sequences are A080998, A081000, A081001, A081028, A081029.

maxTerms = 100; Clear[c, s]; c[n_] := c[n] = Sum[d*EulerPhi[n/d], {d, Divisors[n] }]/n^2; s[terms_] := s[terms] = Sort[Range[terms], c[#1] >= c[#2] & ][[1 ;; maxTerms]]; s[terms = maxTerms]; s[terms += maxTerms]; While[s[terms] != s[terms - maxTerms], terms += maxTerms]; A080997 = s[terms] (* Jean-François Alcover, Feb 19 2015 *)

Formula for centrality of n: A018804(n)/n^2, where A018804(n) is the sum of gcd (k, n) for 1 <= k <= n.

The centrality of a(n) is given by A080999(n)/(a(n))^2.

A199084 a(n) = Sum_{k=1..n} (-1)^(k+1) gcd(k,n).

Original entry on oeis.org

1, -1, 3, -4, 5, -5, 7, -12, 9, -9, 11, -20, 13, -13, 15, -32, 17, -21, 19, -36, 21, -21, 23, -60, 25, -25, 27, -52, 29, -45, 31, -80, 33, -33, 35, -84, 37, -37, 39, -108, 41, -65, 43, -84, 45, -45, 47, -160, 49, -65, 51, -100, 53, -81, 55, -156, 57

Offset: 1

R. J. Mathar, Nov 02 2011

The alternating sum analog of A018804.

a(2n) <= -(2n-1) (cf. A344372). - Max Alekseyev, May 16 2021

Vincenzo Librandi and Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1000 terms from Vincenzo Librandi)
Laszlo Toth, Weighted Gcd-Sum Functions, J. Int. Seq. 14 (2011) # 11.7.7.

Cf. A001620, A018804, A106475, A306016, A344371, A344372, A344373.

A199084 := proc(n)
        add((-1)^(k-1)* igcd(k,n),k=1..n) ;
end proc:
seq(A199084(n),n=1..80) ;

altGCDSum[n_] := Sum[(-1)^(i + 1)GCD[i, n], {i, n}]; Table[altGCDSum[n], {n, 50}] (* Alonso del Arte, Nov 02 2011 *)
Total/@Table[(-1)^(k+1) GCD[k,n],{n,60},{k,n}] (* Harvey P. Dale, May 29 2013 *)

a(n) = sum(k=1, n, (-1)^(k+1)*gcd(k,n)); \\ Michel Marcus, Jun 28 2023

a(2n+1) = 2n+1. - Seiichi Manyama, Dec 09 2016
a(n) = (-1)^(n+1)*A344371(n) = A344373(n) - (-1)^n*n. - Max Alekseyev, May 16 2021
a(2n) = -A344372(n). - Max Alekseyev, May 16 2021
Sum_{k=1..n} a(k) ~ (n^2/Pi^2) * (-log(n) - 2*gamma + 1/2 + 4*log(2)/3 + Pi^2/4 + zeta'(2)/zeta(2)), where gamma is Euler's constant (A001620). - Amiram Eldar, Mar 30 2024

A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

Original entry on oeis.org

1, 1, 2, 7, 8, 14, 32, 121, 126, 248, 512, 1003, 2048, 4064, 8176, 130539, 32768, 65382, 131072, 261868, 524224, 1048064, 2097152, 4193131, 8388576, 16775168, 33554180, 67104688, 134217728, 268426672, 536870912, 8589802359, 2147482624, 4294934528, 8589934336, 17179801257, 34359738368, 68719345664, 137438949376, 274877643724, 549755813888

Offset: 1

Gus Wiseman, Feb 03 2018

Numerators of rational valued sequence f whose Dirichlet convolution with itself yields function g(n) = A000079(n-1) = 2^(n-1). - Antti Karttunen, Aug 10 2018

			Sequence begins: 1, 1, 2, 7/2, 8, 14, 32, 121/2, 126, 248, 512, 1003, 2048, 4064, 8176, 130539/8, 32768.

Andrew Howroyd, Table of n, a(n) for n = 1..1000

Cf. A000005, A000010, A000740, A018804, A023900, A029935, A034691, A046643, A059966, A228369, A257098, A296302, A298971, A299119, A299149, A299152.

Cf. also A317831.

nn=50;
sys=Table[2^(n-1)==Sum[a[d]*a[n/d],{d,Divisors[n]}],{n,nn}];
Numerator[Array[a,nn]/.Solve[sys,Array[a,nn]][[2]]]

A299151perA299152(n) = if(1==n,n,(2^(n-1)-sumdiv(n,d,if((d>1)&&(dA299151perA299152(d)*A299151perA299152(n/d),0)))/2);
A299151(n) = numerator(A299151perA299152(n));

More terms from Antti Karttunen, Jul 29 2018

A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

Original entry on oeis.org

1, 4, 6, 11, 10, 24, 14, 28, 26, 40, 22, 66, 26, 56, 60, 68, 34, 104, 38, 110, 84, 88, 46, 168, 74, 104, 102, 154, 58, 240, 62, 160, 132, 136, 140, 286, 74, 152, 156, 280, 82, 336, 86, 242, 260, 184, 94, 408, 146, 296, 204, 286, 106, 408, 220, 392, 228, 232, 118, 660

Offset: 1

Ilya Gutkovskiy, Aug 29 2019

Dirichlet convolution of Dedekind psi function (A001615) with Euler totient function (A000010).
Dirichlet convolution of A008966 with A018804.
Dirichlet convolution of A038040 with A271102.

Robert Israel, Table of n, a(n) for n = 1..10000
Eric Weisstein's World of Mathematics, Dedekind Function.
Eric Weisstein's World of Mathematics, Totient Function.

Cf. A000010, A001615, A008966, A018804, A038040, A271102.

Cf. A327251 (inverse Möbius transform), A347092 (Dirichlet inverse), A347093 (sum with it), A347135.

f:= proc(n) local t;
  mul((t[2]+1)*t[1]^t[2] - (t[2]-1)*t[1]^(t[2]-2), t = ifactors(n)[2])
end proc:
map(f, [$1..100]); # Robert Israel, Sep 01 2019

Table[Sum[DirichletConvolve[j, MoebiusMu[j]^2, j, n/d] EulerPhi[d], {d, Divisors[n]}], {n, 1, 60}]
f[p_, e_] := (e + 1)*p^e - (e - 1)*p^(e - 2); a[1] = 1; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Oct 26 2020 *)

seq(n) = {dirmul(vector(n, n, eulerphi(n)), vector(n, n, n * sumdivmult(n, d, issquarefree(d)/d)))} \\ Andrew Howroyd, Aug 29 2019

A001615(n) = if(1==n,n, my(f=factor(n)); prod(i=1, #f~, f[i, 1]^f[i, 2] + f[i, 1]^(f[i, 2]-1)));
A322577(n) = sumdiv(n,d,A001615(n/d)*eulerphi(d)); \\ Antti Karttunen, Apr 03 2022

Dirichlet g.f.: zeta(s-1)^2 / zeta(2*s).
a(p) = 2*p, where p is prime.
Sum_{k=1..n} a(k) ~ 45*n^2*(2*Pi^4*log(n) - Pi^4 + 4*gamma*Pi^4 - 360*zeta'(4)) / (2*Pi^8), where gamma is the Euler-Mascheroni constant A001620 and for zeta'(4) see A261506. - Vaclav Kotesovec, Aug 31 2019
a(p^k) = (k+1)*p^k - (k-1)*p^(k-2) where p is prime. - Robert Israel, Sep 01 2019
a(n) = Sum_{k=1..n} psi(gcd(n,k)). - Ridouane Oudra, Nov 29 2019
a(n) = Sum_{k=1..n} psi(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A343499 a(n) = Sum_{k=1..n} gcd(k, n)^5.

Original entry on oeis.org

1, 33, 245, 1058, 3129, 8085, 16813, 33860, 59541, 103257, 161061, 259210, 371305, 554829, 766605, 1083528, 1419873, 1964853, 2476117, 3310482, 4119185, 5315013, 6436365, 8295700, 9778145, 12253065, 14468481, 17788154, 20511177, 25297965, 28629181, 34672912, 39459945, 46855809

Offset: 1

Seiichi Manyama, Apr 17 2021

Seiichi Manyama, Table of n, a(n) for n = 1..10000

Column 5 of A343510.

Cf. A000010, A001159 (sigma_4(n)), A018804, A054612, A059378, A069097, A332517, A342423, A342433, A343497, A343498, A344524.

A343499:= func< n | (&+[d^5*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343499(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^5, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^5);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^5);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 4));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+26*x^k+66*x^(2*k)+26*x^(3*k)+x^(4*k))/(1-x^k)^6))

def A343499(n): return sum(k^5*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343499(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^5.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_4(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 26*x^k + 66*x^(2*k) + 26*x^(3*k) + x^(4*k))/(1 - x^k)^6.
Dirichlet g.f.: zeta(s-1) * zeta(s-5) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 315*zeta(5)*n^6 / (2*Pi^6). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(4*e+5) - p^(4*e) - p + 1)/(p^4-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i_1, ..., i_5 <= n} gcd(i_1, ..., i_5, n) = Sum_{d divides n} d * J_5(n/d), where the Jordan totient function J_5(n) = A059378(n). - Peter Bala, Jan 29 2024

A368736 a(n) = Sum_{k = 1..n} gcd(2*k+1, n).

Original entry on oeis.org

1, 2, 5, 4, 9, 10, 13, 8, 21, 18, 21, 20, 25, 26, 45, 16, 33, 42, 37, 36, 65, 42, 45, 40, 65, 50, 81, 52, 57, 90, 61, 32, 105, 66, 117, 84, 73, 74, 125, 72, 81, 130, 85, 84, 189, 90, 93, 80, 133, 130, 165, 100, 105, 162, 189, 104, 185, 114, 117, 180, 121, 122, 273, 64, 225, 210, 133, 132, 225, 234

Offset: 1

Peter Bala, Jan 04 2024

CF. A000010, A018804, A344372, A368737 - A368742.

seq(add(gcd(2*k+1, n), k = 1..n), n = 1..70);
# alternative faster program for large n
with(numtheory): seq(add(irem(d,2)*phi(d)*n/d, d in divisors(n)), n = 1..70);

Table[Sum[GCD[2*k + 1, n], {k, 1, n}], {n, 1, 100}] (* Vaclav Kotesovec, Jan 11 2024 *)

a(n) = sum(k = 1, n, gcd(2*k+1, n)); \\ Michel Marcus, Jan 11 2024

a(2*n) = 2*a(n); a(2*n+1) = A018804(2*n+1) = A344372(2*n+1).
a(n) = Sum_{k = 1..n} gcd(4*k+1, n) = Sum_{k = 1..n} gcd(4*k+3, n).
a(n) = Sum_{d divides n} X(d)*phi(d)*n/d, where phi(n) = A000010(n) and X(n) = 1 if n is odd, else 0, that is, X(n) is the principal Dirichlet character of the reduced residue system mod 2. See A000035.
Multiplicative: a(2^k) = 2^k and for odd prime p, a(p^e) = (e + 1)*p^e - e*p^(e-1).
Dirichlet g.f.: (1 - 2/2^s)/(1 - 1/2^s) * zeta(s-1)^2/zeta(s).
Sum_{k=1..n} a(k) ~ n^2*(2*log(n) - 1 + 4*gamma + 4*log(2)/3 - 12*zeta'(2)/Pi^2) / Pi^2, where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Jan 11 2024

A130054 Inverse Moebius transform of A023900.

Original entry on oeis.org

1, 0, -1, -1, -3, 0, -5, -2, -3, 0, -9, 1, -11, 0, 3, -3, -15, 0, -17, 3, 5, 0, -21, 2, -7, 0, -5, 5, -27, 0, -29, -4, 9, 0, 15, 3, -35, 0, 11, 6, -39, 0, -41, 9, 9, 0, -45, 3, -11, 0, 15, 11, -51, 0, 27, 10, 17, 0, -57, -3, -59, 0, 15, -5, 33, 0, -65, 15, 21

Offset: 1

Gary W. Adamson, May 04 2007

Multiplicative because A023900 is. - Andrew Howroyd, Aug 03 2018

Amiram Eldar, Table of n, a(n) for n = 1..10000 (terms 1..1000 from Andrew Howroyd)

Row sums of A129691.

Cf. A126988, A000005, A023900, A062570, A001511, A018804, A000203, A007431.

[&+[d*MoebiusMu(d)*NumberOfDivisors(n div d):d in Divisors(n)]:n in [1..70]]; // Marius A. Burtea, Nov 17 2019

with(numtheory): seq(add(d*mobius(d)*tau(n/d), d in divisors(n)), n=1..60); # Ridouane Oudra, Nov 17 2019

b[n_] := Sum[d MoebiusMu[d], {d, Divisors[n]}];
a[n_] := Sum[b[n/d], {d, Divisors[n]}];
a /@ Range[1, 100] (* Jean-François Alcover, Sep 20 2019, from PARI *)
f[p_, e_] := 1-(p-1)*e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 23 2020 *)

\\ here b(n) is A023900
b(n)={sumdivmult(n, d, d*moebius(d))}
a(n)={sumdiv(n, d, b(n/d))} \\ Andrew Howroyd, Aug 03 2018

A126988 * A130054 = d(n), A000005: (1, 2, 2, 3, 2, 4, 2, 4, 3, 4, ...).
a(n) = Sum_{d|n} A023900(n/d). - Andrew Howroyd, Aug 03 2018
a(n) = Sum_{d|n} d*mu(d)*tau(n/d). - Ridouane Oudra, Nov 17 2019
From Werner Schulte, Sep 06 2020: (Start)
Multiplicative with a(p^e) = 1 - (p-1) * e for prime p and e >= 0.
Dirichlet g.f.: (zeta(s))^2 / zeta(s-1).
Dirichlet convolution with A062570 equals A001511.
Dirichlet convolution with A018804 equals A000203.
Dirichlet inverse of A007431. (End)
a(n) = 1 - Sum_{k=1..n-1} a(gcd(n,k)). - Ilya Gutkovskiy, Nov 06 2020

Name changed and terms a(11) and beyond from Andrew Howroyd, Aug 03 2018

cp's OEIS Frontend

A124647 a(n) = (2n + 1)*3^n.

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A067911 Product of gcd(k,n) for 1 <= k <= n.

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A078430 Sum of gcd(k^2,n) for 1 <= k <= n.

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A080997 The positive integers arranged in nonincreasing order of centrality (the fraction of n represented by the average gcd of n and the other positive integers).

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A199084 a(n) = Sum_{k=1..n} (-1)^(k+1) gcd(k,n).

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A299151 Numerators of the positive solution to 2^(n-1) = Sum_{d|n} a(d) * a(n/d).

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A322577 a(n) = Sum_{d|n} psi(n/d) * phi(d).

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