A054618 -id:A054618 - cp's OEIS frontend

A081720 Triangle T(n,k) read by rows, giving number of bracelets (turnover necklaces) with n beads of k colors (n >= 1, 1 <= k <= n).

1, 1, 3, 1, 4, 10, 1, 6, 21, 55, 1, 8, 39, 136, 377, 1, 13, 92, 430, 1505, 4291, 1, 18, 198, 1300, 5895, 20646, 60028, 1, 30, 498, 4435, 25395, 107331, 365260, 1058058, 1, 46, 1219, 15084, 110085, 563786, 2250311, 7472984, 21552969, 1, 78, 3210, 53764, 493131, 3037314

Offset: 1

N. J. A. Sloane, based on information supplied by Gary W. Adamson, Apr 05 2003

From Petros Hadjicostas, Nov 29 2017: (Start)
The formula given below is clear from the programs given in the Maple and Mathematica sections, while the g.f. for column k can be obtained using standard techniques.
If we differentiate the column k g.f. m times, then we can get a formula for row m. (For this sequence, we only need to use this row m formula for 1 <= k <= m, but it is valid even for k>m.) For example, to get the formula for row 8, we have T(n=8,k) = d^8/dx^8 (column k g.f.)/8! evaluated at x=0. Here, "d^8/dx^8" means "8th derivative w.r.t. x" of the column k g.f. Doing so, we get T(n=8, k) = (k^6 - k^5 + k^4 + 3*k^3 + 2*k^2 - 2*k + 4)*(k + 1)*k/16, which is the formula given for sequence A060560. (Here, we use this formula only for 1 <= k <= 8.)
(End)

			1;                                                (A000027)
1,  3;                                            (A000217)
1,  4,  10;                                       (A000292)
1,  6,  21,   55;                                 (A002817)
1,  8,  39,  136,   377;                          (A060446)
1, 13,  92,  430,  1505,   4291;                  (A027670)
1, 18, 198, 1300,  5895,  20646,  60028;          (A060532)
1, 30, 498, 4435, 25395, 107331, 365260, 1058058; (A060560)
...
For example, when n=k=3, we have the following T(3,3)=10 bracelets of 3 beads using up to 3 colors: 000, 001, 002, 011, 012, 022, 111, 112, 122, and 222. (Note that 012 = 120 = 201 = 210 = 102 = 021.) _Petros Hadjicostas_, Nov 29 2017

N. Zagaglia Salvi, Ordered partitions and colourings of cycles and necklaces, Bull. Inst. Combin. Appl., 27 (1999), 37-40.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
Yi Hu, Numerical Transfer Matrix Method of Next-nearest-neighbor Ising Models, Master's Thesis, Duke Univ. (2021).
Yi Hu and Patrick Charbonneau, Numerical transfer matrix study of frustrated next-nearest-neighbor Ising models on square lattices, arXiv:2106.08442 [cond-mat.stat-mech], 2021, cites the 4th column.

Cf. A321791 (extension to n >= 0, k >= 0).

Cf. A081721 (diagonal), A081722 (row sums), column sequences k=2..6: A000029, A027671, A032275, A032276, A056341.

A081720 := proc(n, k)
    local d, t1;
    t1 := 0;
    if n mod 2 = 0 then
        for d from 1 to n do
            if n mod d = 0 then
                t1 := t1+numtheory[phi](d)*k^(n/d);
            end if;
        end do:
        (t1+(n/2)*(1+k)*k^(n/2)) /(2*n) ;
    else
        for d from 1 to n do
            if n mod d = 0 then
                t1 := t1+numtheory[phi](d)*k^(n/d);
            end if;
        end do;
        (t1+n*k^((n+1)/2)) /(2*n) ;
    end if;
end proc:
seq(seq(A081720(n,k),k=1..n),n=1..10) ;

t[n_, k_] := (For[t1 = 0; d = 1, d <= n, d++, If[Mod[n, d] == 0, t1 = t1 + EulerPhi[d]*k^(n/d)]]; If[EvenQ[n], (t1 + (n/2)*(1 + k)*k^(n/2))/(2*n), (t1 + n*k^((n + 1)/2))/(2*n)]); Table[t[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Sep 13 2012, after Maple, updated Nov 02 2017 *)
Needs["Combinatorica`"]; Table[Table[NumberOfNecklaces[n,k,Dihedral],{k,1,n}],{n,1,8}]//Grid  (* Geoffrey Critzer, Oct 07 2012, after code by T. D. Noe in A027671 *)

See Maple code.
From Petros Hadjicostas, Nov 29 2017: (Start)
T(n,k) = ((1+k)*k^{n/2}/2 + (1/n)*Sum_{d|n} phi(n/d)*k^d)/2, if n is even, and = (k^{(n+1)/2} + (1/n)*Sum_{d|n} phi(n/d)*k^d)/2, if n is odd.
G.f. for column k: (1/2)*((k*x+k*(k+1)*x^2/2)/(1-k*x^2) - Sum_{n>=1} (phi(n)/n)*log(1-k*x^n)) provided we chop off the Taylor expansion starting at x^k (and ignore all the terms x^n with n(End)
2*n*T(n,k) = A054618(n,k)+n*(1+k)^(n/2)/2 if n even, = A054618(n,k)+n*k^((n+1)/2) if n odd. - R. J. Mathar, Jan 23 2022

Name edited by Petros Hadjicostas, Nov 29 2017

A228640 a(n) = Sum_{d|n} phi(d)*n^(n/d).

Original entry on oeis.org

0, 1, 6, 33, 280, 3145, 46956, 823585, 16781472, 387422001, 10000100440, 285311670721, 8916103479504, 302875106592409, 11112006930972780, 437893890382391745, 18446744078004651136, 827240261886336764449, 39346408075494964903956, 1978419655660313589124321

Offset: 0

Alois P. Heinz, Aug 28 2013

nonn

Alois P. Heinz, Table of n, a(n) for n = 0..200

Main diagonal of A054618, A054619, A185651.

Cf. A000010, A056665.

[0] cat [&+[EulerPhi(d)*n^(n div d): d in Divisors(n)]:n in [1..20]]; // Marius A. Burtea, Feb 15 2020

with(numtheory):
a:= n-> add(phi(d)*n^(n/d), d=divisors(n)):
seq(a(n), n=0..20);

a[0] = 0; a[n_] := DivisorSum[n, EulerPhi[#]*n^(n/#)&]; Table[a[n], {n, 0, 20}] (* Jean-François Alcover, Mar 21 2017 *)

a(n) = if (n, sumdiv(n, d, eulerphi(d)*n^(n/d)), 0); \\ Michel Marcus, Feb 15 2020; corrected Jun 13 2022

a(n) = sum(k=1, n, n^gcd(k, n)); \\ Seiichi Manyama, Mar 10 2021

from sympy import totient, divisors
def A228640(n):
    return sum(totient(d)*n**(n//d) for d in divisors(n,generator=True)) # Chai Wah Wu, Feb 15 2020

a(n) = Sum_{k=1..n} n^gcd(k,n) = n * A056665(n). - Seiichi Manyama, Mar 10 2021

a(n) = Sum_{k=1..n} n^(n/gcd(n,k))*phi(gcd(n,k))/phi(n/gcd(n,k)). - Richard L. Ollerton, May 07 2021

A054630 T(n,k) = Sum_{d|k} phi(d)*n^(k/d)/k, triangle read by rows, T(n,k) for n >= 1 and 1 <= k <= n.

Original entry on oeis.org

1, 2, 3, 3, 6, 11, 4, 10, 24, 70, 5, 15, 45, 165, 629, 6, 21, 76, 336, 1560, 7826, 7, 28, 119, 616, 3367, 19684, 117655, 8, 36, 176, 1044, 6560, 43800, 299600, 2097684, 9, 45, 249, 1665, 11817, 88725, 683289, 5381685, 43046889, 10, 55, 340, 2530, 20008, 166870, 1428580, 12501280, 111111340, 1000010044

Offset: 1

N. J. A. Sloane, Apr 16 2000, revised Mar 21 2007

T(n, k) is the number of n-ary necklaces of length k (see Ruskey, Savage and Wang). - Peter Luschny, Aug 12 2012, comment corrected at the suggestion of Petros Hadjicostas, Peter Luschny, Sep 10 2018
From Petros Hadjicostas, Sep 12 2018: (Start)
The programs by Peter Luschny below can generate all n-ary necklaces of length k (and all k-ary necklaces of length n) for any positive integer values of n and k, not just for 1 <= k <= n.
From the examples below, we see that the number of 4-ary necklaces of length 3 equals the number of 3-ary necklaces of length 4. The question is whether there are other pairs (n, k) of distinct positive integers such that the number of n-ary necklaces of length k equals the number of k-ary necklaces of length n.
(End)

			Triangle starts:
  1;
  2,  3;
  3,  6, 11;
  4, 10, 24, 70;
  5, 15, 45, 165,  629;
  6, 21, 76, 336, 1560, 7826;
The 24 necklaces over {0,1,2} of length 4 are:
  0000,0001,0002,0011,0012,0021,0022,0101,0102,0111,0112,0121,
  0122,0202,0211,0212,0221,0222,1111,1112,1122,1212,1222,2222.
The 24 necklaces over {0,1,2,3} of length 3 are:
  000,001,002,003,011,012,013,021,022,023,031,032,
  033,111,112,113,122,123,132,133,222,223,233,333.

D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, Addison-Wesley, 2005.

Peter Luschny, Rows 1..45, flattened
H. Fredricksen and I. J. Kessler, An algorithm for generating necklaces of beads in two colors, Discrete Math. 61 (1986), 181-188.
H. Fredricksen and J. Maiorana, Necklaces of beads in k colors and k-ary de Bruijn sequences, Discrete Math. 23(3) (1978), 207-210. Reviewed in MR0523071 (80e:05007).
Peter Luschny, Implementation of the FKM algorithm in SageMath and Julia.
F. Ruskey, C. Savage, and T. M. Y. Wang, Generating necklaces, Journal of Algorithms, 13(3), 1992, 414-430.
Index entries for sequences related to necklaces

Cf. A054631, A054618, A054619, A056665, A215474. Upper triangle of A075195.

A054630(n::Int, k::Int) = div(sum(n^gcd(i,k) for i in 1:k), k)
for n in 1:6
    println([A054630(n, k) for k in 1:n])
end # Peter Luschny, Sep 10 2018

T := (n,k) -> add(n^igcd(i,k), i=1..k)/k:
seq(seq(T(n,k), k=1..n), n=1..10); # Peter Luschny, Sep 10 2018

T[n_, k_] := 1/k Sum[EulerPhi[d] n^(k/d), {d, Divisors[k]}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 30 2018 *)

def A054630(n,k): return (1/k)*add(euler_phi(d)*n^(k/d) for d in divisors(k))
for n in (1..9):
    print([A054630(n,k) for k in (1..n)]) # Peter Luschny, Aug 12 2012

T(n,n) = A056665(n). - Peter Luschny, Aug 12 2012

T(n,k) = (1/k)*Sum_{i=1..k} n^gcd(i, k). - Peter Luschny, Sep 10 2018

A054631 Triangle read by rows: row n (n >= 1) contains the numbers T(n,k) = Sum_{d|n} phi(d)*k^(n/d)/n, for k=1..n.

Original entry on oeis.org

1, 1, 3, 1, 4, 11, 1, 6, 24, 70, 1, 8, 51, 208, 629, 1, 14, 130, 700, 2635, 7826, 1, 20, 315, 2344, 11165, 39996, 117655, 1, 36, 834, 8230, 48915, 210126, 720916, 2097684, 1, 60, 2195, 29144, 217045, 1119796, 4483815, 14913200, 43046889

Offset: 1

N. J. A. Sloane, Apr 16 2000, revised Mar 21 2007

T(n,k) is the number of n-bead necklaces with up to k different colored beads. - Yves-Loic Martin, Sep 29 2020

			1;
1,  3;                                   (A000217)
1,  4,  11;                              (A006527)
1,  6,  24,   70;                        (A006528)
1,  8,  51,  208,   629;                 (A054620)
1, 14, 130,  700,  2635,  7826;          (A006565)
1, 20, 315, 2344, 11165, 39996, 117655;  (A054621)

Seiichi Manyama, Rows n = 1..140, flattened
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A054630, A054618, A054619, A087854. Lower triangle of A075195.

A054631 := proc(n,k) add( numtheory[phi](d)*k^(n/d),d=numtheory[divisors](n) ) ;  %/n ; end proc: # R. J. Mathar, Aug 30 2011

Needs["Combinatorica`"]; Table[Table[NumberOfNecklaces[n, k, Cyclic], {k, 1, n}], {n, 1, 8}] //Grid (* Geoffrey Critzer, Oct 07 2012, after code by T. D. Noe in A027671 *)
t[n_, k_] := Sum[EulerPhi[d]*k^(n/d)/n, {d, Divisors[n]}]; Table[t[n, k], {n, 1, 9}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jan 20 2014 *)

T(n, k) = sumdiv(n, d, eulerphi(d)*k^(n/d))/n; \\ Seiichi Manyama, Mar 10 2021

T(n, k) = sum(j=1, n, k^gcd(j, n))/n; \\ Seiichi Manyama, Mar 10 2021

T(n,k) = Sum_{j=1..k} binomial(k,j) * A087854(n, j). - Yves-Loic Martin, Sep 29 2020

T(n,k) = (1/n) * Sum_{j=1..n} k^gcd(j, n). - Seiichi Manyama, Mar 10 2021

A054619 Triangle T(n,k) = Sum_{d|k} phi(d)*n^(k/d).

Original entry on oeis.org

1, 2, 6, 3, 12, 33, 4, 20, 72, 280, 5, 30, 135, 660, 3145, 6, 42, 228, 1344, 7800, 46956, 7, 56, 357, 2464, 16835, 118104, 823585, 8, 72, 528, 4176, 32800, 262800, 2097200, 16781472, 9, 90, 747, 6660, 59085, 532350, 4783023, 43053480, 387422001

Offset: 1

N. J. A. Sloane, Apr 16 2000

			1;
2, 6;
3, 12, 33;
4, 20, 72,  280;
5, 30, 135, 660,  3145;
6, 42, 228, 1344, 7800, 46956;
...

Alois P. Heinz, Rows n = 1..141, flattened

Cf. A054618, A054630, A054631, A185651 (transpose).

Main diagonal gives: A228640.

with(numtheory):
T:= (n, k)-> add(phi(d)*n^(k/d), d=divisors(k)):
seq(seq(T(n,k), k=1..n), n=1..10);  # Alois P. Heinz, Aug 28 2013

T[n_, k_] := Sum[EulerPhi[d]*n^(k/d), {d, Divisors[k]}]; Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Feb 25 2015 *)

T(n, k) = sumdiv(k, d, eulerphi(d)*n^(k/d)); \\ Michel Marcus, Feb 25 2015

Showing 1-5 of 5 results.

cp's OEIS Frontend

A081720 Triangle T(n,k) read by rows, giving number of bracelets (turnover necklaces) with n beads of k colors (n >= 1, 1 <= k <= n).

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A228640 a(n) = Sum_{d|n} phi(d)*n^(n/d).

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A054630 T(n,k) = Sum_{d|k} phi(d)*n^(k/d)/k, triangle read by rows, T(n,k) for n >= 1 and 1 <= k <= n.

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A054631 Triangle read by rows: row n (n >= 1) contains the numbers T(n,k) = Sum_{d|n} phi(d)*k^(n/d)/n, for k=1..n.

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A054619 Triangle T(n,k) = Sum_{d|k} phi(d)*n^(k/d).

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