A069097 -id:A069097 - cp's OEIS frontend

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

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)

A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 1, 0, 1, 4, 0, 0, 0, 1, 2, 2, 1, 0, 0, 1, 6, 0, 0, 0, 0, 0, 1, 4, 2, 0, 1, 0, 0, 0, 1, 6, 0, 2, 0, 0, 0, 0, 0, 1, 4, 4, 0, 0, 1, 0, 0, 0, 0, 1, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1, 12, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 6, 6

Offset: 1

N. J. A. Sloane, Apr 09 2000

From Gary W. Adamson, Jan 08 2007: (Start)
Let H be this lower triangular matrix. Then:
H * A051731 = A126988,
H * [1, 2, 3, ...] = 1, 3, 5, 8, 9, 15, ... = A018804,
H * sigma(n) = A038040 = d(n) * n = 1, 4, 6, 12, 10, ... where sigma(n) = A000203,
H * d(n) (A000005) = sigma(n) = A000203,
Row sums are A000027 (corrected by Werner Schulte, Sep 06 2020, see comment of Gary W. Adamson, Aug 03 2008),
H^2 * d(n) = d(n)*n, H^2 = A127192,
H * mu(n) (A008683) = A007431(n) (corrected by Werner Schulte, Sep 06 2020),
H^2 row sums = A018804. (End)
The Möbius inversion principle of Richard Dedekind and Joseph Liouville (1857), cf. "Concrete Mathematics", p. 136, is equivalent to the statement that row sums are the row index n. - Gary W. Adamson, Aug 03 2008
The multivariable row polynomials give n times the cycle index for the cyclic group C_n, called Z(C_n) (see the MathWorld link with the Harary reference): n*Z(C_n) = Sum_{k=1..n} T(n,k)*(y_{n/k})^k, n >= 1. E.g., 6*Z(C_6) = 2*(y_6)^1 + 2*(y_3)^2 + 1*(y_2)^3 + 1*(y_1)^6. - Wolfdieter Lang, May 22 2012
See A102190 (no 0's, rows reversed). - Wolfdieter Lang, May 29 2012
This is the number of permutations in the n-th cyclic group which are the product of k disjoint cycles. - Robert A. Beeler, Aug 09 2013

			Triangle begins
   1;
   1, 1;
   2, 0, 1;
   2, 1, 0, 1;
   4, 0, 0, 0, 1;
   2, 2, 1, 0, 0, 1;
   6, 0, 0, 0, 0, 0, 1;
   4, 2, 0, 1, 0, 0, 0, 1;
   6, 0, 2, 0, 0, 0, 0, 0, 1;
   4, 4, 0, 0, 1, 0, 0, 0, 0, 1;
  10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1;
   4, 2, 2, 2, 0, 1, 0, 0, 0, 0, 0, 1;

Ronald L. Graham, D. E. Knuth, Oren Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., 1994, p. 136.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
Eric Weisstein's World of Mathematics, Cycle Index.

Cf. A000005, A000007, A000010, A000012, A000203, A007431, A008683.
Cf. A038040, A051731, A054521, A054525, A102190, A126988.
Sums incliude: A029935, A069097, A092843 (diagonal), A209295.
Sums of the form Sum_{k} k^p * T(n, k): A000027 (p=0), A018804 (p=1), A069097 (p=2), A343497 (p=3), A343498 (p=4), A343499 (p=5).

a054523 n k = a054523_tabl !! (n-1) !! (k-1)
a054523_row n = a054523_tabl !! (n-1)
a054523_tabl = map (map (\x -> if x == 0 then 0 else a000010 x)) a126988_tabl
-- Reinhard Zumkeller, Jan 20 2014

A054523:= func< n,k | k eq n select 1 else (n mod k) eq 0 select EulerPhi(Floor(n/k)) else 0 >;
[A054523(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Jun 24 2024

A054523 := proc(n,k) if n mod k = 0 then numtheory[phi](n/k) ; else 0; end if; end proc: # R. J. Mathar, Apr 11 2011

T[n_, k_]:= If[k==n,1,If[Divisible[n, k], EulerPhi[n/k], 0]];
Table[T[n,k], {n,15}, {k,n}]//Flatten (* G. C. Greubel, Dec 15 2017 *)

for(n=1, 10, for(k=1, n, print1(if(!(n % k), eulerphi(n/k), 0), ", "))) \\ G. C. Greubel, Dec 15 2017

def A054523(n,k):
    if (k==n): return 1
    elif (n%k)==0: return euler_phi(int(n//k))
    else: return 0
flatten([[A054523(n,k) for k in range(1,n+1)] for n in range(1,16)]) # G. C. Greubel, Jun 24 2024

Sum_{k=1..n} k * T(n, k) = A018804(n). - Gary W. Adamson, Jan 08 2007
Equals A054525 * A126988 as infinite lower triangular matrices. - Gary W. Adamson, Aug 03 2008
From Werner Schulte, Sep 06 2020: (Start)
Sum_{k=1..n} T(n,k) * A000010(k) = A029935(n) for n > 0.
Sum_{k=1..n} k^2 * T(n,k) = A069097(n) for n > 0. (End)
From G. C. Greubel, Jun 24 2024: (Start)
T(2*n-1, n) = A000007(n-1), n >= 1.
T(2*n, n) = A000012(n), n >= 1.
Sum_{k=1..n} (-1)^(k-1)*T(n, k) = (1 - (-1)^n)*n/2.
Sum_{k=1..floor(n+1)/2} T(n-k+1, k) = A092843(n+1).
Sum_{k=1..n} (k+1)*T(n, k) = A209295(n).
Sum_{k=1..n} k^3 * T(n, k) = A343497(n).
Sum_{k=1..n} k^4 * T(n, k) = A343498(n).
Sum_{k=1..n} k^5 * T(n, k) = A343499(n). (End)

A332517 a(n) = Sum_{k=1..n} gcd(n,k)^n.

Original entry on oeis.org

1, 5, 29, 274, 3129, 47515, 823549, 16843268, 387459861, 10009769725, 285311670621, 8918311856102, 302875106592265, 11112685048729175, 437893951473411261, 18447025557276459016, 827240261886336764193, 39346558373052524325225, 1978419655660313589123997

Offset: 1

Ilya Gutkovskiy, Feb 14 2020

nonn

If n is prime, a(n) = n-1 + n^n. - Robert Israel, Feb 16 2020

Robert Israel, Table of n, a(n) for n = 1..386

Cf. A000010, A008683, A018804, A031971, A069097, A226561, A228640, A321294.

[&+[Gcd(n,k)^n:k in [1..n]]: n in [1..20]]; // Marius A. Burtea, Feb 15 2020

f:= n -> add(igcd(n,k)^n,k=1..n):
map(f, [$1..30]); # Robert Israel, Feb 16 2020

Table[Sum[GCD[n, k]^n, {k, 1, n}], {n, 1, 19}]
Table[Sum[EulerPhi[n/d] d^n, {d, Divisors[n]}], {n, 1, 19}]
Table[Sum[MoebiusMu[n/d] d DivisorSigma[n - 1, d], {d, Divisors[n]}], {n, 1, 19}]

a(n) = sum(k=1, n, gcd(n, k)^n); \\ Michel Marcus, Feb 14 2020

from sympy import totient, divisors
def A332517(n):
    return sum(totient(d)*(n//d)**n for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = Sum_{d|n} phi(n/d) * d^n.
a(n) = Sum_{d|n} mu(n/d) * d * sigma_(n-1)(d).
a(n) ~ n^n.
From Richard L. Ollerton, May 09 2021: (Start)
a(n) = Sum_{k=1..n} (n/gcd(n,k))^n*phi(gcd(n,k))/phi(n/gcd(n,k)).
a(n) = Sum_{k=1..n} mu(n/gcd(n,k))*gcd(n,k)*sigma_(n-1)(gcd(n,k))/phi(n/gcd(n,k)). (End)

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

A343510 Square array T(n,k), n >= 1, k >= 1, read by antidiagonals, where T(n,k) = Sum_{j=1..n} gcd(j, n)^k.

Original entry on oeis.org

1, 1, 3, 1, 5, 5, 1, 9, 11, 8, 1, 17, 29, 22, 9, 1, 33, 83, 74, 29, 15, 1, 65, 245, 274, 129, 55, 13, 1, 129, 731, 1058, 629, 261, 55, 20, 1, 257, 2189, 4162, 3129, 1411, 349, 92, 21, 1, 513, 6563, 16514, 15629, 8085, 2407, 596, 105, 27, 1, 1025, 19685, 65794, 78129, 47515, 16813, 4388, 789, 145, 21

Offset: 1

Seiichi Manyama, Apr 17 2021

			G.f. of column 3: Sum_{i>=1} phi(i) * (x^i + 4*x^(2*i) + x^(3*i))/(1 - x^i)^4.
Square array begins:
   1,  1,   1,    1,     1,      1,      1, ...
   3,  5,   9,   17,    33,     65,    129, ...
   5, 11,  29,   83,   245,    731,   2189, ...
   8, 22,  74,  274,  1058,   4162,  16514, ...
   9, 29, 129,  629,  3129,  15629,  78129, ...
  15, 55, 261, 1411,  8085,  47515, 282381, ...
  13, 55, 349, 2407, 16813, 117655, 823549, ...

Seiichi Manyama, Antidiagonals n = 1..140, flattened
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.
Eric Weisstein's World of Mathematics, Eulerian Number and Euler's Number Triangle

Columns k=1..7 give A018804, A069097, A343497, A343498, A343499, A343508, A343509.
T(n-2,n) gives A342432.
T(n-1,n) gives A342433.
T(n,n) gives A332517.
T(n,n+1) gives A321294.
Cf. A008292, A343516.

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

PARI
```
T(n, k) = sum(j=1, n, gcd(j, n)^k);
```

T(n, k) = sumdiv(n, d, eulerphi(n/d)*d^k);

T(n, k) = sumdiv(n, d, moebius(n/d)*d*sigma(d, k-1));

G.f. of column k: Sum_{i>=1} phi(i) * ( Sum_{j=1..k} A008292(k, j) * x^(i*j) )/(1 - x^i)^(k+1).
T(n,k) = Sum_{d|n} phi(n/d) * d^k.
T(n,k) = Sum_{d|n} mu(n/d) * d * sigma_{k-1}(d).
Dirichlet g.f. of column k: zeta(s-1) * zeta(s-k) / zeta(s). - Ilya Gutkovskiy, Apr 18 2021
T(n,k) = Sum_{j=1..n} (n/gcd(n,j))^k*phi(gcd(n,j))/phi(n/gcd(n,j)). - Richard L. Ollerton, May 10 2021
T(n,k) = Sum_{1 <= j_1, j_2, ..., j_k <= n} gcd(j_1, j_2, ..., j_k)^2 = Sum_{d divides n} d * J_k(n/d), where J_k(n) denotes the k-th Jordan totient function. - Peter Bala, Jan 29 2024

A283123 a(n) = sigma(9*n).

Original entry on oeis.org

13, 39, 40, 91, 78, 120, 104, 195, 121, 234, 156, 280, 182, 312, 240, 403, 234, 363, 260, 546, 320, 468, 312, 600, 403, 546, 364, 728, 390, 720, 416, 819, 480, 702, 624, 847, 494, 780, 560, 1170, 546, 960, 572, 1092, 726, 936, 624, 1240, 741, 1209

Offset: 1

Seiichi Manyama, Mar 01 2017

nonn

In general, for k>=1, Sum_{j=1..n} sigma(j*k) ~ A069097(k) * Pi^2 * n^2 / (12*k). - Vaclav Kotesovec, May 11 2024

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

Sigma(k*n): A000203 (k=1), A062731 (k=2), A144613 (k=3), A193553 (k=4), A283118 (k=5), A224613 (k=6), A283078 (k=7), A283122 (k=8), this sequence (k=9).

Cf. A008591.

Table[DivisorSigma[1, 9 n], {n, 50}] (* Michael De Vlieger, Mar 01 2017 *)

a(n) = sigma(9*n) \\ Amiram Eldar, Dec 16 2022

a(n) = A000203(9*n).

Sum_{k=1..n} a(k) = (35*Pi^2/36) * n^2 + O(n*log(n)). - Amiram Eldar, Dec 16 2022

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

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

A368743 a(n) = Sum_{1 <= i, j <= n} gcd(i, j, n)^3.

Original entry on oeis.org

1, 11, 35, 100, 149, 385, 391, 848, 1017, 1639, 1451, 3500, 2365, 4301, 5215, 6976, 5201, 11187, 7219, 14900, 13685, 15961, 12695, 29680, 19225, 26015, 28107, 39100, 25229, 57365, 30751, 56576, 50785, 57211, 58259, 101700, 52021, 79409, 82775, 126352

Offset: 1

Peter Bala, Jan 20 2024

Amiram Eldar, Table of n, a(n) for n = 1..10000
Peter Bala, GCD sum theorems. Two Multivariable Cesaro Type Identities.

Cf. A007434, A059376, A069097, A343497, A343498, A360428.

seq( add(add(igcd(i, j, n)^3, i = 1..n), j = 1..n), n = 1..50);
# faster program for large n
with(numtheory):
A007434 := proc(n) add(d^2*mobius(n/d), d in divisors(n)) end proc:
seq( add(d^3*A007434(n/d), d in divisors(n)), n = 1..500);

f[p_, e_] := p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Jan 29 2024 *)

a(n) = {my(f = factor(n)); prod(i = 1, #f~, p = f[i,1]; e = f[i,2]; p^(3*e - 2)*(p^2 + p + 1) - p^(2*e - 2)*(p + 1));} \\ Amiram Eldar, Jan 29 2024

from math import prod
from sympy import factorint
def A368743(n): return prod(p**(e-1<<1)*(p**e*(p*(q:=p+1)+1)-q) for p, e in factorint(n).items()) # Chai Wah Wu, Jan 29 2024

a(n) = Sum_{1 <= i, j, k <= n} gcd(i, j, k, n)^2.
a(n) = Sum_{d divides n} d^3 * J_2(n/d) = Sum_{d divides n} d^2 * J_3(n/d), where the Jordan totient functions J_2(n) = A007434(n) and J_3(n) = A059376(n).
Dirichlet g.f.: zeta(s-2) * zeta(s-3)/zeta(s).
a(n) is a multiplicative function: for prime p, a(p^k) = p^(3*k-2)*(p^2 + p + 1) - p^(2*k-2)*(p + 1).
Sum_{k=1..n} a(k) ~ c * n^4, where c = 15/(4*Pi^2) = 0.379954... . - Amiram Eldar, Jan 29 2024
a(n) = Sum_{d divides n} mobius(n/d) * d^2 * sigma(d). - Peter Bala, Jan 29 2024

cp's OEIS Frontend

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

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A054523 Triangle read by rows: T(n,k) = phi(n/k) if k divides n, T(n,k)=0 otherwise (n >= 1, 1 <= k <= n).

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A332517 a(n) = Sum_{k=1..n} gcd(n,k)^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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