A343510 -id:A343510 - cp's OEIS frontend

A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

1, 3, 5, 8, 9, 15, 13, 20, 21, 27, 21, 40, 25, 39, 45, 48, 33, 63, 37, 72, 65, 63, 45, 100, 65, 75, 81, 104, 57, 135, 61, 112, 105, 99, 117, 168, 73, 111, 125, 180, 81, 195, 85, 168, 189, 135, 93, 240, 133, 195, 165, 200, 105, 243, 189, 260, 185, 171, 117, 360

Offset: 1

David W. Wilson

a(n) is the number of times the number 1 appears in the character table of the cyclic group C_n. - Ahmed Fares (ahmedfares(AT)my-deja.com), Jun 02 2001
a(n) is the number of ways to express all fractions f/g whereby each product (f/g)*n is a natural number between 1 and n (using fractions of the form f/g with 1 <= f,g <= n). For example, for n=4 there are 8 such fractions: 1/1, 1/2, 2/2, 3/3, 1/4, 2/4, 3/4 and 4/4. - Ron Lalonde (ronronronlalonde(AT)hotmail.com), Oct 03 2002
Number of non-congruent solutions to xy == 0 (mod n). - Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 06 2003
Conjecture: n>1 divides a(n)+1 iff n is prime. - Thomas Ordowski, Oct 22 2014
The above conjecture is false, with counterexample given by n = 3*37*43*42307*116341 and a(n)+1 = 26*n. - Varun Vejalla, Jun 19 2025
a(n) is the number of 0's in the multiplication table Z/nZ (cf. A000010 for number of 1's). - Eric Desbiaux, Jun 11 2015
{a(n)} == 1, 3, 1, 0, 1, 3, 1, 0, ... (mod 4). - Isaac Saffold, Dec 30 2017
Since a(p^e) = p^(e-1)*((p-1)e+p) it follows a(p) = 2p-1 and therefore p divides a(p)+1. - Ruediger Jehn, Jun 23 2022

			G.f. = x + 3*x^2 + 5*x^3 + 8*x^4 + 9*x^5 + 15*x^6 + 13*x^7 + 20*x^8 + ...

S. S. Pillai, On an arithmetic function, J. Annamalai University 2 (1933), pp. 243-248.
J. Sándor, A generalized Pillai function, Octogon Mathematical Magazine Vol. 9, No. 2 (2001), 746-748.

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 2000 terms from T. D. Noe)
U. Abel, W. Awan, and V. Kushnirevych, A Generalization of the Gcd-Sum Function, J. Int. Seq. 16 (2013) #13.6.7.
Antal Bege, Hadamard product of GCD matrices, Acta Univ. Sapientiae, Mathematica, 1, 1 (2009) 43-49.
Olivier Bordelles, The Composition of the gcd and Certain Arithmetic Functions, J. Int. Seq. 13 (2010) #10.7.1.
Olivier Bordelles, An Asymptotic Formula for Short Sums of Gcd-Sum Functions, Journal of Integer Sequences, Vol. 15 (2012), #12.6.8.
Olivier Bordelles, A Multidimensional Cesáro Type Identity and Applications, J. Int. Seq. 18 (2015) # 15.3.7.
Kevin A. Broughan, The gcd-sum function, Journal of Integer Sequences 4 (2001), Article 01.2.2, 19 pp.
J.-M. De Koninck and I. Kátai, Some remarks on a paper of L. Toth, JIS 13 (2010) 10.1.2.
Pentti Haukkanen and László Tóth, Menon-type identities again: Note on a paper by Li, Kim and Qiao, arXiv:1911.05411 [math.NT], 2019.
Soichi Ikeda and Kaneaki Matsuoka, On the Lcm-Sum Function, Journal of Integer Sequences, Vol. 17 (2014), Article 14.1.7.
Mathematical Reflections, Solution to Problem O364, Issue 2, 2016, p 24. [Cached copy at Wayback Machine]
Taylor McAdam, Almost-primes in horospherical flows on the space of lattices, arXiv:1802.08764 [math.DS], 2018.
Taylor McAdam, Almost-prime times in horospherical flows on the space of lattices, Journal of Modern Dynamics (2019) Vol. 15, 277-327.
J. Ransford, A Sum of Gcd’s over Friable Numbers, JIS vol 19 (2016) # 16.3.2
Jeffrey Shallit, Problem E 2821, American Mathematical Monthly 87 (1980), 220.
Jeffrey Shallit, Solution, American Mathematical Monthly, 88 (1981), 444-445.
László Tóth, A gcd-sum function over regular integers modulo n, JIS 12 (2009) 09.2.5.
László Tóth, On the Bi-Unitary Analogues of Euler's Arithmetical Function and the Gcd-Sum Function, JIS 12 (2009) 09.5.2.
László Tóth, A survey of gcd-sum functions, J. Integer Sequences 13 (2010), Article 10.8.1, 23 pp.
László Tóth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7.
D. Zhang and W. Zhai, Mean Values of a Gcd-Sum Function Over Regular Integers Modulo n, J. Int. Seq. 13 (2010), 10.4.7.
D. Zhang and W. Zhai, Mean Values of a Class of Arithmetical Functions, J. Int. Seq. 14 (2011) #11.6.5.
D. Zhang and W. Zhai, On an Open Problem of Tóth, J. Int. Seq. 16 (2013) #13.6.5.

Column 1 of A343510 and A343516.
Cf. A080997, A080998 for rankings of the positive integers in terms of centrality, defined to be the average fraction of an integer that it shares with the other integers as a gcd, or A018804(n)/n^2, also A080999, a permutation of this sequence (A080999(n) = A018804(A080997(n))).
Cf. A185210, A010766, A054523, A127468, A050873, A078430, A095026, A227507, A000010, A344372, A272718 (partial sums), A368736 - A368741.

a018804 n = sum $ map (gcd n) [1..n]  -- Reinhard Zumkeller, Jul 16 2012

[&+[Gcd(n,k):k in [1..n]]:n in [1..60]]; // Marius A. Burtea, Nov 14 2019

a:=n->sum(igcd(n,j),j=1..n): seq(a(n), n=1..60); # Zerinvary Lajos, Nov 05 2006

f[n_] := Block[{d = Divisors[n]}, Sum[ d*EulerPhi[n/d], {d, d}]]; Table[f[n], {n, 60}] (* Robert G. Wilson v, Mar 20 2012 *)
a[ n_] := If[ n < 1, 0, n Sum[ EulerPhi[d] / d, {d, Divisors@n}]]; (* Michael Somos, Jan 07 2017 *)
f[p_, e_] := (e*(p - 1)/p + 1)*p^e; a[n_] := Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Jul 19 2019 *)

{a(n) = direuler(p=2, n, (1 - X) / (1 - p*X)^2)[n]}; /* Michael Somos, May 31 2000 */

a(n)={ my(ct=0); for(i=0,n-1,for(j=0,n-1, ct+=(Mod(i*j,n)==0) ) ); ct; } \\ Joerg Arndt, Aug 03 2013

a(n)=my(f=factor(n)); prod(i=1,#f~,(f[i,2]*(f[i,1]-1)/f[i,1] + 1)*f[i,1]^f[i,2]) \\ Charles R Greathouse IV, Oct 28 2014

a(n) = sumdiv(n, d, n*eulerphi(d)/d); \\ Michel Marcus, Jan 07 2017

from sympy.ntheory import totient, divisors
print([sum(n*totient(d)//d for d in divisors(n)) for n in range(1, 101)]) # Indranil Ghosh, Apr 04 2017

from sympy import factorint
from math import prod
def A018804(n): return prod(p**(e-1)*((p-1)*e+p) for p, e in factorint(n).items()) # Chai Wah Wu, Nov 29 2021

a(n) = Sum_{d|n} d*phi(n/d), where phi(n) is Euler totient function (cf. A000010). - Vladeta Jovovic, Apr 04 2001
Multiplicative; for prime p, a(p^e) = p^(e-1)*((p-1)e+p).
Dirichlet g.f.: zeta(s-1)^2/zeta(s).
a(n) = Sum_{d|n} d*tau(d)*mu(n/d). - Benoit Cloitre, Oct 23 2003
Equals A054523 * [1,2,3,...]. Equals row sums of triangle A010766. - Gary W. Adamson, May 20 2007
Equals inverse Mobius transform of A029935 = A054525 * (1, 2, 4, 5, 8, 8, 12, 12, ...). - Gary W. Adamson, Aug 02 2008, corrected Feb 07 2023
Equals row sums of triangle A127478. - Gary W. Adamson, Aug 03 2008
G.f.: Sum_{k>=1} phi(k)*x^k/(1 - x^k)^2, where phi(k) is the Euler totient function. - Ilya Gutkovskiy, Jan 02 2017
a(n) = Sum_{a = 1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1, for n > 1. The sum is over a,b,c such that n*c - a*b = 0. - Benedict W. J. Irwin, Apr 04 2017
Proof: Let gcd(a, n) = g and x = n/g. Define B = {x, 2*x, ..., g*x}; then for all b in B there exists a number c such that a*b = n*c. Since the set B has g elements it follows that Sum_{b=1..n} Sum_{c=1..n} 1 >= g = gcd(a, n) and therefore Sum_{a=1..n} Sum_{b=1..n} Sum_{c=1..n} 1 >= Sum_{a=1..n} gcd(a, n). On the other hand, for all b not in B there is no number c <= n such that a*b = n*c and hence Sum_{b = 1..n} Sum_{c = 1..n} 1 = g. Therefore Sum_{a=1..n} Sum_{b = 1..n} Sum_{c = 1..n} 1 = a(n). - Ruediger Jehn, Jun 23 2022
a(2*n) = a(n)*(3-A007814(n)/(A007814(n)+2)) - Velin Yanev, Jun 30 2017
Proof: Let m = A007814(m) and decompose n into n = k*2^m. We know from Chai Wah Wu's program below that a(n) = Product(p_i^(e_i-1)*((p_i-1)*e_i+p_i)) where the numbers p_i are the prime factors of n and e_i are the corresponding exponents. Hence a(2n) = 2^m*(m+3)*a(k) = 2^m*(m+3)*a(k). On the other hand, a(n) = 2^(m-1)*(m+2)*a(k). Dividing the first equation by the second yields a(2n)/a(n) = 2*(m+3)/(m+2), which equals 3 - m/(m+2). Hence a(2n) = a(n)*(3 - m/(m+2)). - Ruediger Jehn, Jun 23 2022
Sum_{k=1..n} a(k) ~ 3*n^2/Pi^2 * (log(n) - 1/2 + 2*gamma - 6*Zeta'(2)/Pi^2), where gamma is the Euler-Mascheroni constant A001620. - Vaclav Kotesovec, Feb 08 2019
a(n) = Sum_{k=1..n} n/gcd(n,k)*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 10 2021
log(a(n)/n) << log n log log log n/log log n; in particular, a(n) << n^(1+e) for any e > 0. See Broughan link for bounds in terms of omega(n). - Charles R Greathouse IV, Sep 08 2022
a(n) = (1/4)*Sum_{k = 1..4*n} (-1)^k * gcd(k, 4*n) = (1/4) * A344372(2*n). - Peter Bala, Jan 01 2024

A069097 Moebius transform of A064987, n*sigma(n).

Original entry on oeis.org

1, 5, 11, 22, 29, 55, 55, 92, 105, 145, 131, 242, 181, 275, 319, 376, 305, 525, 379, 638, 605, 655, 551, 1012, 745, 905, 963, 1210, 869, 1595, 991, 1520, 1441, 1525, 1595, 2310, 1405, 1895, 1991, 2668, 1721, 3025, 1891, 2882, 3045, 2755, 2255, 4136, 2737

Offset: 1

Benoit Cloitre, Apr 05 2002

Equals A127569 * [1, 2, 3, ...]. - Gary W. Adamson, Jan 19 2007
Equals row sums of triangle A143309 and of triangle A143312. - Gary W. Adamson, Aug 06 2008
Dirichlet convolution of A000290 and A000010 (see Jovovic formula). - R. J. Mathar, Feb 03 2011

Amiram Eldar, Table of n, a(n) for n = 1..10000
Wenchang Chu, Jordan's totient function and trigonometric sums, Studia Scientiarum Mathematicarum Hungarica 57 (1), 40-53 (2020).
Nikolai Osipov, Problem 12003, Amer. Math. Monthly, 124(8) (2017), p. 754.

Column 2 of A343510.
Cf. A018804, A033457, A068963, A064987, A127569, A143309, A143312, A360428.
For Sum_{k = 1..n} gcd(k,n)^m see A018804 (m = 1), A343497 (m = 3), A343498 (m = 4) and A343499 (m = 5).

A069097[n_]:=n^2*Plus @@((EulerPhi[#]/#^2)&/@ Divisors[n]); Array[A069097, 100] (* Enrique Pérez Herrero, Feb 25 2012 *)
f[p_, e_] := p^(e-1)*(p^e*(p+1)-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 18 2020 *)

for(n=1,100,print1((sumdiv(n,k,k*sigma(k)*moebius(n/k))),","))

a(n) = Sum_{d|n} d^2*phi(n/d). - Vladeta Jovovic, Jul 31 2002
a(n) = Sum_{k=1..n} gcd(n, k)^2. - Vladeta Jovovic, Aug 27 2003
Dirichlet g.f.: zeta(s-2)*zeta(s-1)/zeta(s). - R. J. Mathar, Feb 03 2011
a(n) = n*Sum_{d|n} J_2(d)/d, where J_2 is A007434. - Enrique Pérez Herrero, Feb 25 2012.
G.f.: Sum_{n >= 1} phi(n)*(x^n + x^(2*n))/(1 - x^n)^3 = x + 5*x^2 + 11*x^3 + 22*x^4  + .... - Peter Bala, Dec 30 2013
Multiplicative with a(p^e) = p^(e-1)*(p^e*(p+1)-1). - R. J. Mathar, Jun 23 2018
Sum_{k=1..n} a(k) ~ Pi^2 * n^3 / (18*zeta(3)). - Vaclav Kotesovec, Sep 18 2020
a(n) = Sum_{k=1..n} (n/gcd(n,k))^2*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021
From Peter Bala, Dec 26 2023: (Start)
For n odd, a(n) = Sum_{k = 1..n} gcd(k,n)/cos(k*Pi/n)^2 (see Osipov and also Chu, p. 51).
It appears that for n odd, Sum_{k = 1..n} (-1)^(k+1)*gcd(k,n)/cos(k*Pi/n)^2 = n. (End)
a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n). Cf. A360428. - Peter Bala, Jan 16 2024
Sum_{k=1..n} a(k)/k ~ Pi^2 * n^2 / (12*zeta(3)). - Vaclav Kotesovec, May 11 2024

A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

Original entry on oeis.org

1, 9, 29, 74, 129, 261, 349, 596, 789, 1161, 1341, 2146, 2209, 3141, 3741, 4776, 4929, 7101, 6877, 9546, 10121, 12069, 12189, 17284, 16145, 19881, 21321, 25826, 24417, 33669, 29821, 38224, 38889, 44361, 45021, 58386, 50689, 61893, 64061, 76884, 68961, 91089, 79549, 99234, 101781

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 3 of A343510.

Cf. A000010, A001157 (sigma_2(n)), A018804, A054610, A069097, A309323, A332517, A342423, A342433, A343498, A343499, A343513.

A343497:= func< n | (&+[d^3*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343497(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

with(numtheory):
seq(add(phi(n/d) * d^3, d in divisors(n)), n = 1..50); # Peter Bala, Jan 20 2024

a[n_] := Sum[GCD[k, n]^3, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e - 1)*((p^2 + p + 1)*p^(2*e) - 1)/(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50] (* Amiram Eldar, Nov 22 2022 *)
A343497[n_]:= DivisorSum[n, #^3*EulerPhi[n/#] &]; Table[A343497[n], {n, 50}] (* G. C. Greubel, Jun 24 2024 *)

PARI
```
a(n) = sum(k=1, n, gcd(k, n)^3);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^3);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 2));

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

def A343497(n): return sum(k^3*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343497(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^3.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_2(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 4*x^k + x^(2*k))/(1 - x^k)^4.
Dirichlet g.f.: zeta(s-1) * zeta(s-3) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ 45*zeta(3)*n^4 / (2*Pi^4). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*((p^2+p+1)*p^(2*e) - 1)/(p+1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n) = Sum_{d divides n} d * J_3(n/d), where the Jordan totient function J_3(n) = A059376(n). - Peter Bala, Jan 20 2024

A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

Original entry on oeis.org

1, 17, 83, 274, 629, 1411, 2407, 4388, 6729, 10693, 14651, 22742, 28573, 40919, 52207, 70216, 83537, 114393, 130339, 172346, 199781, 249067, 279863, 364204, 393145, 485741, 545067, 659518, 707309, 887519, 923551, 1123472, 1216033, 1420129, 1514003, 1843746, 1874197

Offset: 1

Seiichi Manyama, Apr 17 2021

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

Column 4 of A343510.

Cf. A000010, A001158 (sigma_3(n)), A018804, A054611, A069097, A332517, A342423, A342433, A343497, A343499, A343514.

A343498:= func< n | (&+[d^4*EulerPhi(Floor(n/d)): d in Divisors(n)]) >;
[A343498(n): n in [1..50]]; // G. C. Greubel, Jun 24 2024

a[n_] := Sum[GCD[k, n]^4, {k, 1, n}]; Array[a, 50] (* Amiram Eldar, Apr 18 2021 *)
f[p_, e_] := p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1); 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)^4);
```

a(n) = sumdiv(n, d, eulerphi(n/d)*d^4);

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, 3));

my(N=40, x='x+O('x^N)); Vec(sum(k=1, N, eulerphi(k)*x^k*(1+11*x^k+11*x^(2*k)+x^(3*k))/(1-x^k)^5))

def A343498(n): return sum(k^4*euler_phi(n/k) for k in (1..n) if (k).divides(n))
[A343498(n) for n in range(1,51)] # G. C. Greubel, Jun 24 2024

a(n) = Sum_{d|n} phi(n/d) * d^4.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_3(d).
G.f.: Sum_{k >= 1} phi(k) * x^k * (1 + 11*x^k + 11*x^(2*k) + x^(3*k))/(1 - x^k)^5.
Dirichlet g.f.: zeta(s-1) * zeta(s-4) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
Sum_{k=1..n} a(k) ~ Pi^4 * n^5 / (450*zeta(5)). - Vaclav Kotesovec, May 20 2021
Multiplicative with a(p^e) = p^(e-1)*(p^(3*e+4) - p^(3*e) - p + 1)/(p^3-1). - Amiram Eldar, Nov 22 2022
a(n) = Sum_{1 <= i, j, k, l <= n} gcd(i, j, k, l, n) = Sum_{d divides n} d * J_4(n/d), where the Jordan totient function J_4(n) = A059377(n). - Peter Bala, Jan 18 2024

A343516 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ... , x_k, n).

Original entry on oeis.org

1, 1, 3, 1, 4, 5, 1, 5, 8, 8, 1, 6, 12, 15, 9, 1, 7, 17, 26, 19, 15, 1, 8, 23, 42, 39, 35, 13, 1, 9, 30, 64, 74, 76, 34, 20, 1, 10, 38, 93, 130, 153, 90, 56, 21, 1, 11, 47, 130, 214, 287, 216, 152, 63, 27, 1, 12, 57, 176, 334, 506, 468, 379, 191, 86, 21

Offset: 1

Seiichi Manyama, Apr 17 2021

			T(4,2) = gcd(1,1,4) + gcd(1,2,4) + gcd(2,2,4) + gcd(1,3,4) + gcd(2,3,4) + gcd(3,3,4) + gcd(1,4,4) + gcd(2,4,4) + gcd(3,4,4) + gcd(4,4,4) = 1 + 1 + 2 + 1 + 1 + 1 + 1 + 2 + 1 + 4 = 15.
Square array begins:
   1,  1,  1,   1,   1,   1,    1, ...
   3,  4,  5,   6,   7,   8,    9, ...
   5,  8, 12,  17,  23,  30,   38, ...
   8, 15, 26,  42,  64,  93,  130, ...
   9, 19, 39,  74, 130, 214,  334, ...
  15, 35, 76, 153, 287, 506,  846, ...
  13, 34, 90, 216, 468, 930, 1722, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..7 give A018804, A309322, A309323, A343518, A343519, A343520, A343521.
Main diagonal gives A343517.
T(n,n-1) gives A343553.
Cf. A343510.

T[n_, k_] := DivisorSum[n, EulerPhi[n/#] * Binomial[k + # - 1, k] &]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, Apr 18 2021 *)

T(n, k) = sumdiv(n, d, eulerphi(n/d)*binomial(d+k-1, k));

G.f. of column k: Sum_{j>=1} phi(j) * x^j/(1 - x^j)^(k+1).

T(n,k) = Sum_{d|n} phi(n/d) * binomial(d+k-1, k).

A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

Original entry on oeis.org

1, 3, 11, 74, 629, 8085, 117655, 2113796, 43059849, 1001955177, 25937424611, 743379914746, 23298085122493, 793811662313709, 29192938251553759, 1152956691126550536, 48661191875666868497, 2185928270773974154773

Offset: 1

Seiichi Manyama, Mar 12 2021

nonn

Seiichi Manyama, Table of n, a(n) for n = 1..387

Cf. A000010, A000272, A321349, A332517, A332621, A342432, A343510.

a[n_] := Sum[GCD[k, n]^(n - 1), {k, 1, n}]; Array[a, 20] (* Amiram Eldar, Mar 12 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*d^(n-1));

a(n) = sumdiv(n, d, moebius(n/d)*d*sigma(d, n-2));

a(n) = Sum_{d|n} phi(n/d) * d^(n-1).
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-2)(d).
a(n) ~ n^(n-1). - Vaclav Kotesovec, May 23 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

A344479 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1, x_2, ..., x_k <= n} gcd(x_1, x_2, ..., x_k).

Original entry on oeis.org

1, 1, 3, 1, 5, 6, 1, 9, 12, 10, 1, 17, 30, 24, 15, 1, 33, 84, 76, 37, 21, 1, 65, 246, 276, 141, 61, 28, 1, 129, 732, 1060, 649, 267, 80, 36, 1, 257, 2190, 4164, 3165, 1417, 400, 112, 45, 1, 513, 6564, 16516, 15697, 8091, 2528, 624, 145, 55, 1, 1025, 19686, 65796, 78261, 47521, 17128, 4432, 885, 189, 66

Offset: 1

Seiichi Manyama, May 22 2021

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
   1,  1,   1,    1,    1,     1, ...
   3,  5,   9,   17,   33,    65, ...
   6, 12,  30,   84,  246,   732, ...
  10, 24,  76,  276, 1060,  4164, ...
  15, 37, 141,  649, 3165, 15697, ...
  21, 61, 267, 1417, 8091, 47521, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened

Columns k=1..5 give A000217, A018806, A344522, A344523, A344524.
T(n,n) gives A344525.
Cf. A343510, A343516, A344527.

T[n_, k_] := Sum[EulerPhi[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)

T(n, k) = sum(j=1, n, eulerphi(j)*(n\j)^k);

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.

T(n,k) = Sum_{j=1..n} phi(j) * floor(n/j)^k.

A344527 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) is the number of ordered k-tuples (x_1, x_2, ..., x_k) with gcd(x_1, x_2, ..., x_k) = 1 (1 <= {x_1, x_2, ..., x_k} <= n).

Original entry on oeis.org

1, 1, 1, 1, 3, 1, 1, 7, 7, 1, 1, 15, 25, 11, 1, 1, 31, 79, 55, 19, 1, 1, 63, 241, 239, 115, 23, 1, 1, 127, 727, 991, 607, 181, 35, 1, 1, 255, 2185, 4031, 3091, 1199, 307, 43, 1, 1, 511, 6559, 16255, 15559, 7501, 2303, 439, 55, 1, 1, 1023, 19681, 65279, 77995, 45863, 16531, 3823, 637, 63, 1

Offset: 1

Seiichi Manyama, May 22 2021

			G.f. of column 3: (1/(1 - x)) * Sum_{i>=1} mu(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^3.
Square array begins:
  1,  1,   1,    1,    1,     1, ...
  1,  3,   7,   15,   31,    63, ...
  1,  7,  25,   79,  241,   727, ...
  1, 11,  55,  239,  991,  4031, ...
  1, 19, 115,  607, 3091, 15559, ...
  1, 23, 181, 1199, 7501, 45863, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened.

Columns k=1..6 give A000012, A018805, A071778, A082540, A082544, A343978.
T(n,n) gives A332468.
Cf. A343510, A343516, A344479.

T[n_, k_] := Sum[MoebiusMu[j] * Quotient[n, j]^k, {j, 1, n}]; Table[T[k, n - k + 1], {n, 1, 11}, {k, 1, n}] // Flatten (* Amiram Eldar, May 22 2021 *)

T(n, k) = sum(j=1, n, moebius(j)*(n\j)^k);

PARI
```
T(n, k) = n^k-sum(j=2, n, T(n\j, k));
```

from functools import lru_cache
from itertools import count, islice
@lru_cache(maxsize=None)
def A344527_T(n,k):
    if n == 0:
        return 0
    c, j, k1 = 1, 2, n//2
    while k1 > 1:
        j2 = n//k1 + 1
        c += (j2-j)*A344527_T(k1,k)
        j, k1 = j2, n//j2
    return n*(n**(k-1)-1)-c+j
def A344527_gen(): # generator of terms
    return (A344527_T(k+1, n-k) for n in count(1) for k in range(n))
A344527_list = list(islice(A344527_gen(),30)) # Chai Wah Wu, Nov 02 2023

G.f. of column k: (1/(1 - x)) * Sum_{i>=1} mu(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^k.
T(n,k) = Sum_{j=1..n} mu(j) * floor(n/j)^k.
T(n,k) = n^k - Sum_{j=2..n} T(floor(n/j),k).

cp's OEIS Frontend

A018804 Pillai's arithmetical function: Sum_{k=1..n} gcd(k, n).

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A069097 Moebius transform of A064987, n*sigma(n).

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A343497 a(n) = Sum_{k=1..n} gcd(k, n)^3.

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A343498 a(n) = Sum_{k=1..n} gcd(k, n)^4.

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A343516 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{1 <= x_1 <= x_2 <= ... <= x_k <= n} gcd(x_1, x_2, ... , x_k, n).

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A342433 a(n) = Sum_{k=1..n} gcd(k,n)^(n-1).

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