A054612 -id:A054612 - cp's OEIS frontend

A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.

0, 0, 0, 0, 1, 0, 0, 2, 2, 0, 0, 3, 6, 3, 0, 0, 4, 12, 12, 4, 0, 0, 5, 20, 33, 24, 5, 0, 0, 6, 30, 72, 96, 40, 6, 0, 0, 7, 42, 135, 280, 255, 84, 7, 0, 0, 8, 56, 228, 660, 1040, 780, 140, 8, 0, 0, 9, 72, 357, 1344, 3145, 4200, 2205, 288, 9, 0

Offset: 0

Alois P. Heinz, Aug 29 2013

Dirichlet convolution of phi(n) and k^n. - Richard L. Ollerton, May 07 2021

			Square array A(n,k) begins:
  0, 0,  0,   0,    0,     0,     0, ...
  0, 1,  2,   3,    4,     5,     6, ...
  0, 2,  6,  12,   20,    30,    42, ...
  0, 3, 12,  33,   72,   135,   228, ...
  0, 4, 24,  96,  280,   660,  1344, ...
  0, 5, 40, 255, 1040,  3145,  7800, ...
  0, 6, 84, 780, 4200, 15810, 46956, ...

Alois P. Heinz, Antidiagonals n = 0..140, flattened

Columns k=0..10 give A000004, A001477, A053635, A054610, A054611, A054612, A054613, A054614, A054615, A054616, A054617.
Rows n=0..10 give A000004, A001477, A002378, A054602, A054603, A054604, A054605, A054606, A054607, A054608, A054609.
Main diagonal gives A228640.
Cf. A000010, A075195, A258170.

with(numtheory):
A:= (n, k)-> add(phi(d)*k^(n/d), d=divisors(n)):
seq(seq(A(n, d-n), n=0..d), d=0..12);

a[, 0] = a[0, ] = 0; a[n_, k_] := Sum[EulerPhi[d]*k^(n/d), {d, Divisors[n]}]; Table[a[n - k, k], {n, 0, 12}, {k, n, 0, -1}] // Flatten (* Jean-François Alcover, Dec 06 2013 *)

A(n,k) = Sum_{d|n} phi(d)*k^(n/d).
A(n,k) = Sum_{i=0..min(n,k)} C(k,i) * i! * A258170(n,i). - Alois P. Heinz, May 22 2015
G.f. for column k: Sum_{n>=1} phi(n)*k*x^n/(1-k*x^n) for k >= 0. - Petros Hadjicostas, Nov 06 2017
From Richard L. Ollerton, May 07 2021: (Start)
A(n,k) = Sum_{i=1..n} k^gcd(n,i).
A(n,k) = Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))/phi(n/gcd(n,i)).
A(n,k) = A075195(n,k)*n for n >= 1, k >= 1.  (End)

A001869 Number of n-bead necklaces with 5 colors.

Original entry on oeis.org

1, 5, 15, 45, 165, 629, 2635, 11165, 48915, 217045, 976887, 4438925, 20346485, 93900245, 435970995, 2034505661, 9536767665, 44878791365, 211927736135, 1003867701485, 4768372070757, 22706531350485, 108372083629275, 518301258916445

Offset: 0

N. J. A. Sloane

From Richard L. Ollerton, May 07 2021: (Start)
Here, as in A000031, turning over is not allowed.
(1/n) * Dirichlet convolution of phi(n) and 5^n, n>0. (End)

J. Riordan, An Introduction to Combinatorial Analysis, Wiley, 1958, p. 162.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
R. P. Stanley, Enumerative Combinatorics, Cambridge, Vol. 2, 1999; see Problem 7.112(a).

T. D. Noe, Table of n, a(n) for n=0..200
Joscha Diehl, Rosa Preiß, and Jeremy Reizenstein, Conjugation, loop and closure invariants of the iterated-integrals signature, arXiv:2412.19670 [math.RA], 2024. See p. 21.
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.
INRIA Algorithms Project, Encyclopedia of Combinatorial Structures 5
Juhani Karhumäki, S. Puzynina, M. Rao, and M. A. Whiteland, On cardinalities of k-abelian equivalence classes, arXiv preprint arXiv:1605.03319 [math.CO], 2016.
J. Riordan, Letter to N. J. A. Sloane, Jul. 1978
Eric Weisstein's World of Mathematics, Necklace.
Index entries for sequences related to necklaces

Column 5 of A075195.

Cf. A054612.

CoefficientList[Series[1-Sum[EulerPhi[i] Log[1-5*x^i]/i,{i,1,mx}],{x,0,mx}],x] (* Herbert Kociemba, Nov 01 2016 *)
k=5; Prepend[Table[DivisorSum[n, EulerPhi[#] k^(n/#) &]/n, {n, 1, 30}], 1] (* Robert A. Russell, Sep 21 2018 *)

a(n) = if (n, sumdiv(n, d, eulerphi(d)*5^(n/d))/n, 1); \\ Michel Marcus, Nov 01 2016

a(n) = (1/n)*Sum_{d|n} phi(d)*5^(n/d), n > 0.
G.f.: 1 - Sum_{n>=1} phi(n)*log(1 - 5*x^n)/n. - Herbert Kociemba, Nov 01 2016
a(0) = 1; a(n) = (1/n) * Sum_{k=1..n} 5^gcd(n,k). - Ilya Gutkovskiy, Apr 17 2021
a(0) = 1; a(n) = (1/n)*Sum_{k=1..n} 5^(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

A343492 a(n) = Sum_{k=1..n} 5^(gcd(k, n) - 1).

Original entry on oeis.org

1, 6, 27, 132, 629, 3162, 15631, 78264, 390681, 1953774, 9765635, 48831564, 244140637, 1220718786, 6103516983, 30517656528, 152587890641, 762939850086, 3814697265643, 19073488283028, 95367431672037, 476837167968810, 2384185791015647, 11920929004069128

Offset: 1

Seiichi Manyama, Apr 17 2021

nonn

Column 5 of A343489.

Cf. A000010, A054612.

a[n_] := Sum[5^(GCD[k, n] - 1), {k, 1, n}]; Array[a, 24] (* Amiram Eldar, Apr 17 2021 *)

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

a(n) = sumdiv(n, d, eulerphi(n/d)*5^(d-1));

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

a(n) = Sum_{d|n} phi(n/d)*5^(d - 1) = A054612(n)/5.

G.f.: Sum_{k>=1} phi(k) * x^k / (1 - 5*x^k).

cp's OEIS Frontend

A185651 A(n,k) = Sum_{d|n} phi(d)*k^(n/d); square array A(n,k), n>=0, k>=0, read by antidiagonals.

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A001869 Number of n-bead necklaces with 5 colors.

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

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A343492 a(n) = Sum_{k=1..n} 5^(gcd(k, n) - 1).

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