A215474 -id:A215474 - cp's OEIS frontend

A060223 Number of orbits of length n under the map whose periodic points are counted by A000670.

1, 1, 1, 4, 18, 108, 778, 6756, 68220, 787472, 10224702, 147512052, 2340963570, 40527565260, 760095923082, 15352212731820, 332228417589720, 7668868648772700, 188085259069430744, 4884294069438337428, 133884389812214097774, 3863086904690670182596

Offset: 0

Thomas Ward, Mar 21 2001

From Gus Wiseman, Oct 14 2016: (Start)
A finite sequence is normal if it spans an initial interval of positive integers. The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling the sequences together. For example, (2 2 1) * (2 1 3) = (2 1 2 2 1 3). If Q is the set of compositions (finite sequences of positive integers) then (Q,*) is an Abelian group freely generated by a set P of prime sequences. The number of normal prime sequences of length n is equal to a(n). See example 2 and Mathematica program 2.
If N is the species (endofunctor over the category of finite sets and permutations) of unlabeled necklaces and N(S) represents the set of all non-isomorphic primitive necklaces of length n=|S|, then the numbers |N(S)| are equal to the numbers a(|S|) for any finite set S. This is because the number of orderless *-factorizations (see A034691 and A269134) of any finite sequence q is equal to the number of multiset partitions (see A007716 and A255906) of the multiset of prime factors of q. (End)

			a(5) = 108 since A000670(5) is 541 and A000670(1) is 1, so there must be (541-1)/5 = 108 orbits of length 5.
From _Gus Wiseman_, Oct 14 2016: (Start)
The a(4) = 18 normal prime sequences are the columns:
[2 2 2 3 3 3 3 3 3 3 3 3 4 4 4 4 4 4]
[1 2 2 1 1 1 2 2 2 2 3 3 1 1 2 2 3 3]
[1 1 2 1 2 2 1 1 2 3 1 2 2 3 1 3 1 2]
[1 1 1 2 1 2 1 2 1 1 2 1 3 2 3 1 2 1].
The symmetric function A(x_1,x_2,x_3,...) expanded in terms of monomial symmetric functions m(y) (indexed by integer partitions y) is equal to:
A = m(1) +
    m(11) +
    (2*m(21) + 2*m(111) +
    (m(22) + 2*m(31) + 9*m(211) + 6*m(1111)) +
    (4*m(32) + 2*m(41) + 18*m(221) + 12*m(311) + 48*m(2111) + 24*m(11111)) +
    (3*m(33) + 4*m(42) + 2*m(51) + 14*m(222) + 60*m(321) + 15*m(411) + 180*m(2211) + 80*m(3111) + 300*m(21111) + 120*m(111111)) + ... (End)

Alois P. Heinz, Table of n, a(n) for n = 0..200
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
T. Ward, Exactly realizable sequences

Cf. A000670, A034691 (multisets of compositions), A269134, A007716, A277427, A215474, A255906.

Row sums of A254040.

a[n_] := DivisorSum[n, MoebiusMu[#] HurwitzLerchPhi[1/2, -n/#, 0]/2 &] / n; a[0] = 1; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 30 2016 *)
thufbin[{},b_List]:=b;thufbin[a_List,{}]:=a;thufbin[a_List]:=a;
thufbin[{x_,a___},{y_,b___}]:=Switch[Ordering[If[x=!=y,{x,y},{thufbin[{a},{x,b}],thufbin[{x,a},{b}]}]],{1,2},Prepend[thufbin[{a},{y,b}],x],{2,1},Prepend[thufbin[{x,a},{b}],y]];
thufbin[a_List,b_List,c__List]:=thufbin[a,thufbin[b,c]];
priseqs[n_]:=Fold[Select,Tuples[Range[n],n],{Union[#]===Range[First[#]]&,Function[q,Select[Table[List[Take[q,{1,j}],Take[q,{j+1,n}]],{j,1,n-1}],thufbin@@Sort[#]===q&,1]==={}]}];
Table[Length[priseqs[n]],{n,1,7}] (* Gus Wiseman, Oct 14 2016 *)

\\ here b(n) is A000670
b(n)={polcoeff(serlaplace(1/(2-exp(x+O(x*x^n)))), n)}
a(n)={if(n<1, n==0, sumdiv(n, d, moebius(d)*b(n/d))/n)} \\ Andrew Howroyd, Dec 12 2017

a(n) = (1/n)* Sum_{d|n} mu(d)*A000670(n/d) for n > 0, where mu is A008683, the Moebius function. - Edited by Michel Marcus, Mar 30 2016
Let A = Sum_{q in P} Prod_i x_{q_i} = Sum_y c_y m(y) be the symmetric function whose coefficient of m(y) is equal to the number of permutations of the normal multiset [k]^y that belong to P, where the multiplicity of i in [k]^y is defined to be y_i. Then a(n) is the sum of c_y taken over all integer partitions of n. See example 3. - Gus Wiseman, Oct 14 2016
a(n) = Sum_{d|n} mu(d) * A019536(n/d) for n >= 1. - Petros Hadjicostas, Aug 19 2019

More terms from Alois P. Heinz, Jan 23 2015

A074650 Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n >= 1, k >= 1.

Original entry on oeis.org

1, 2, 0, 3, 1, 0, 4, 3, 2, 0, 5, 6, 8, 3, 0, 6, 10, 20, 18, 6, 0, 7, 15, 40, 60, 48, 9, 0, 8, 21, 70, 150, 204, 116, 18, 0, 9, 28, 112, 315, 624, 670, 312, 30, 0, 10, 36, 168, 588, 1554, 2580, 2340, 810, 56, 0, 11, 45, 240, 1008, 3360, 7735, 11160, 8160, 2184, 99, 0

Offset: 1

Christian G. Bower, Aug 28 2002

D. E. Knuth uses the term 'prime strings' for Lyndon words because of the fundamental theorem stating the unique factorization of strings into nonincreasing prime strings (see Knuth 7.2.1.1). With this terminology T(n,k) is the number of k-ary n-tuples (a_1,...,a_n) such that the string a_1...a_n is prime. - Peter Luschny, Aug 14 2012
Also, for k a power of a prime, the number of monic irreducible polynomials of degree n over GF(k). - Andrew Howroyd, Dec 23 2017
An equivalent description: Array read by antidiagonals: T(n,k) = number of conjugacy classes of primitive words of length k >= 1 over an alphabet of size n >= 1.
There are a few incorrect values in Table 1 in the Perrin-Reutenauer paper (Christophe Reutenauer, personal communication), see A294438. - Lars Blomberg, Dec 05 2017
The fact that T(3,4) = 20 coincides with the number of the amino acids encoded by DNA made Francis Crick, John Griffith and Leslie Orgel conjecture in 1957 that the genetic code is a comma-free code, which later turned out to be false. [Hayes] - Andrey Zabolotskiy, Mar 24 2018

			T(4, 3) counts the 18 ternary prime strings of length 4 which are: 0001, 0002, 0011, 0012, 0021, 0022, 0102, 0111, 0112, 0121, 0122, 0211, 0212, 0221, 0222, 1112, 1122, 1222.
Square array starts:
  1,  2,   3,    4,     5,     6,      7, ...
  0,  1,   3,    6,    10,    15,     21, ...
  0,  2,   8,   20,    40,    70,    112, ...
  0,  3,  18,   60,   150,   315,    588, ...
  0,  6,  48,  204,   624,  1554,   3360, ...
  0,  9, 116,  670,  2580,  7735,  19544, ...
  0, 18, 312, 2340, 11160, 39990, 117648, ...
  ...
The transposed array starts:
   1  0  0     0     0      0       0        0         0          0,
   2  1  2     3     6      9      18       30        56         99,
   3  3  8    18    48    116     312      810      2184       5880,
   4  6  20   60   204    670    2340     8160     29120     104754,
   5 10  40  150   624   2580   11160    48750    217000     976248,
   6 15  70  315  1554   7735   39990   209790   1119720    6045837,
   7 21 112  588  3360  19544  117648   720300   4483696   28245840,
   8 28 168 1008  6552  43596  299592  2096640  14913024  107370900,
   9 36 240 1620 11808  88440  683280  5380020  43046640  348672528,
  10 45 330 2475 19998 166485 1428570 12498750 111111000  999989991,
  11 55 440 3630 32208 295020 2783880 26793030 261994040 2593726344,
  12 66 572 5148 49764 497354 5118828 53745120 573308736 6191711526,
  ...
The initial antidiagonals are:
   1
   2  0
   3  1   0
   4  3   2    0
   5  6   8    3    0
   6 10  20   18    6     0
   7 15  40   60   48     9     0
   8 21  70  150  204   116    18     0
   9 28 112  315  624   670   312    30     0
  10 36 168  588 1554  2580  2340   810    56    0
  11 45 240 1008 3360  7735 11160  8160  2184   99   0
  12 55 330 1620 6552 19544 39990 48750 29120 5880 186 0

F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, p. 97 (2.3.74)
Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, p. 495.
D. E. Knuth, Generating All Tuples and Permutations. The Art of Computer Programming, Vol. 4, Fascicle 2, pp. 26-27, Addison-Wesley, 2005.

Alois P. Heinz, Antidiagonals n = 1..141, flattened
B. Hayes, The invention of the genetic code, American Scientist, Vol. 86, No. 1 (January-February 1998), pp. 8-14.
Veronika Irvine, Lace Tessellations: A mathematical model for bobbin lace and an exhaustive combinatorial search for patterns, PhD Dissertation, University of Victoria, 2016.
Irem Kucukoglu and Yilmaz Simsek, On k-ary Lyndon words and their generating functions, AIP Conference Proceedings 1863, 300004 (2017).
R. C. Lyndon, On Burnside's problem, Transactions of the American Mathematical Society 77, (1954) 202-215.
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, arXiv preprint arXiv:1609.05438 [math.CO] (2016).
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243.
Dominique Perrin and Christophe Reutenauer, Hall sets, Lazard sets and comma-free codes, Discrete Math., 341 (2018), 232-243. [Annotated scanned copy of page 236 only.]
Wikipedia, Lyndon word
Index entries for sequences related to Lyndon words

Columns k: A001037 (k=2), A027376 (k=3), A027377 (k=4), A001692 (k=5), A032164 (k=6), A001693 (k=7), A027380 (k=8), A027381 (k=9), A032165 (k=10), A032166 (k=11), A032167 (k=12), A060216 (k=13), A060217 (k=14), A060218 (k=15), A060219 (k=16), A060220 (k=17), A060221 (k=18), A060222 (k=19).
Rows n: A000027 (n=1), A000217(k-1) (n=2), A007290(k+1) (n=3), A006011 (n=4), A208536(k+1) (n=5), A292350 (n=6), A208537(k+1) (n=7).
Cf. A000010, A008683, A075147 (main diagonal), A102659, A215474 (preprime strings), A383011.

t:= func< n,k | (&+[MoebiusMu(Floor(n/d))*k^d: d in Divisors(n)])/n >; // array
A074650:= func< n,k | t(k, n-k+1) >; // downward diagonals
[A074650(n,k): k in [1..n], n in [1..15]]; // G. C. Greubel, Aug 01 2024

with(numtheory):
T:= proc(n, k) add(mobius(n/d)*k^d, d=divisors(n))/n end:
seq(seq(T(i, 1+d-i), i=1..d), d=1..11);  # Alois P. Heinz, Mar 28 2008

max = 12; t[n_, k_] := Total[ MoebiusMu[n/#]*k^# & /@ Divisors[n]]/n; Flatten[ Table[ t[n-k+1, k], {n, 1, max}, {k, n, 1, -1}]] (* Jean-François Alcover, Oct 18 2011, after Maple *)

T(n,k)=sumdiv(n,d,moebius(n/d)*k^d)/n \\ Charles R Greathouse IV, Oct 18 2011

# This algorithm generates and counts all k-ary n-tuples (a_1,..,a_n) such
# that the string a_1...a_n is prime. It is algorithm F in Knuth 7.2.1.1.
def A074650(n, k):
    a = [0]*(n+1); a[0]=-1
    j = 1; count = 0
    while(j != 0) :
        if j == n : count += 1; # print("".join(map(str,a[1:])))
        else: j = n
        while a[j] >= k-1 : j -= 1
        a[j] += 1
        for i in (j+1..n): a[i] = a[i-j]
    return count   # Peter Luschny, Aug 14 2012

T(n,k) = (1/n) * Sum_{d|n} mu(n/d)*k^d.
T(n,k) = (k^n - Sum_{dAlois P. Heinz, Mar 28 2008
From Richard L. Ollerton, May 10 2021: (Start)
T(n,k) = (1/n)*Sum_{i=1..n} mu(gcd(n,i))*k^(n/gcd(n,i))/phi(n/gcd(n,i)).
T(n,k) = (1/n)*Sum_{i=1..n} mu(n/gcd(n,i))*k^gcd(n,i)/phi(n/gcd(n,i)). (End)
From Seiichi Manyama, Apr 12 2025: (Start)
G.f. of column k: -Sum_{j>=1} mu(j) * log(1 - k*x^j) / j.
Product_{n>=1} 1/(1 - x^n)^T(n,k) = 1/(1 - k*x). (End)

A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.

Original entry on oeis.org

1, 2, 2, 1, 3, 2, 1, 1, 3, 1, 4, 2, 1, 1, 1, 2, 2, 1, 3, 1, 1, 3, 2, 4, 1, 5, 2, 1, 1, 1, 1, 2, 2, 1, 1, 3, 1, 1, 1, 3, 1, 2, 3, 2, 1, 4, 1, 1, 4, 2, 5, 1, 6, 2, 1, 1, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 1, 2, 2, 2, 1, 3, 1, 1, 1, 1, 3, 1, 1, 2, 3, 1, 2, 1, 3, 2, 1, 1, 3, 2, 2, 3, 3, 1, 4, 1, 1, 1, 4, 1, 2, 4, 2, 1, 4, 3, 5, 1, 1, 5, 2, 6, 1, 7

Offset: 1

Gus Wiseman, Jan 12 2017

The *-product of two or more finite sequences is defined to be the lexicographically minimal sequence obtainable by shuffling them together. Every finite positive integer sequence has a unique *-factorization using prime compositions P = {(1), (2), (21), (3), (211), ...}. See A060223 and A228369 for details.

These are co-Lyndon compositions, ordered first by sum and then lexicographically. - Gus Wiseman, Nov 15 2019

			The prime factorization of (1, 1, 4, 2, 3, 1, 5, 5) is: (11423155) = (1)*(1)*(5)*(5)*(4231). The prime factorizations of the initial terms of A000002 are:
             (1) = (1)
            (12) = (1)*(2)
           (122) = (1)*(2)*(2)
          (1221) = (1)*(221)
         (12211) = (1)*(2211)
        (122112) = (1)*(2)*(2211)
       (1221121) = (1)*(221121)
      (12211212) = (1)*(2)*(221121)
     (122112122) = (1)*(2)*(2)*(221121)
    (1221121221) = (1)*(221)*(221121)
   (12211212212) = (1)*(2)*(221)*(221121)
  (122112122122) = (1)*(2)*(2)*(221)*(221121).
Read as a sequence:
(1), (2), (21), (3), (211), (31), (4), (2111), (221), (311), (32), (41), (5).
Read as a triangle:
(1)
(2)
(21), (3)
(211), (31), (4)
(2111), (221), (311), (32), (41), (5).
Read as a sequence of triangles:
1    2    2 1    2 1 1    2 1 1 1    2 1 1 1 1    2 1 1 1 1 1
          3      3 1      2 2 1      2 2 1 1      2 1 2 1 1
                 4        3 1 1      3 1 1 1      2 2 1 1 1
                          3 2        3 1 2        2 2 2 1
                          4 1        3 2 1        3 1 1 1 1
                          5          4 1 1        3 1 1 2
                                     4 2          3 1 2 1
                                     5 1          3 2 1 1
                                     6            3 2 2
                                                  3 3 1
                                                  4 1 1 1
                                                  4 1 2
                                                  4 2 1
                                                  4 3
                                                  5 1 1
                                                  5 2
                                                  6 1
                                                  7.

Cf. A000740, A215474, A228369, A277427.
The binary version is A329318.
The binary non-"co" version is A102659.
A sequence listing all Lyndon compositions is A294859.
Numbers whose binary expansion is co-Lyndon are A328596.
Numbers whose binary expansion is co-Lyndon are A275692.
Binary Lyndon words are A001037.
Lyndon compositions are A059966.
Normal Lyndon words are A060223.
Cf. A211097, A211100, A296372, A296373, A298941, A329131, A329312, A329313, A329314, A329324, A329326.

colynQ[q_]:=Array[Union[{RotateRight[q,#],q}]=={RotateRight[q,#],q}&,Length[q]-1,1,And];
lexsort[f_,c_]:=OrderedQ[PadRight[{f,c}]];
Table[Sort[Select[Join@@Permutations/@IntegerPartitions[n],colynQ],lexsort],{n,5}] (* Gus Wiseman, Nov 15 2019 *)

Row lengths are A059966(n) = number of prime compositions of n.

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

A277427 Prime permutations, ordered lexicographically.

Original entry on oeis.org

1, 2, 1, 3, 1, 2, 3, 2, 1, 4, 1, 2, 3, 4, 1, 3, 2, 4, 2, 1, 3, 4, 2, 3, 1, 4, 3, 1, 2, 4, 3, 2, 1, 5, 1, 2, 3, 4, 5, 1, 2, 4, 3, 5, 1, 3, 2, 4, 5, 1, 3, 4, 2, 5, 1, 4, 2, 3, 5, 1, 4, 3, 2, 5, 2, 1, 3, 4, 5, 2, 1, 4, 3, 5, 2, 3, 1, 4, 5, 2, 3, 4, 1, 5, 2, 4, 1, 3, 5, 2, 4, 3, 1, 5, 3, 1, 2, 4, 5, 3, 1, 4, 2, 5, 3, 2, 1

Offset: 1

Gus Wiseman, Oct 14 2016

A permutation of {1..n} is prime (in the sense of A215474) iff it is of the form (n, q_1, q_2, ..., q_{n-1}).

Row n in the triangle consists of all permutations consisting of n followed by a permutation of 1..n-1, in lexicographic order.

			The sequence of prime permutations begins:
1,
21,
312, 321,
4123, 4132, 4213, 4231, 4312, 4321,
...

Gus Wiseman, Table of n, a(n) for n = 1..5912

Cf. A215474, A254040, A060223, A005117.

seq(op(map(t -> (n,op(t)), combinat:-permute(n-1))), n=1..6); # Robert Israel, Nov 07 2016

row[n_] := Join[{n}, #]& /@ Permutations[Range[n-1]];
Array[row, 5] // Flatten (* Jean-François Alcover, Apr 10 2019 *)

A215475 Number of n-ary n-tuples (a_1,...,a_n) such that the string a_1...a_n is preprime.

Original entry on oeis.org

1, 3, 14, 90, 829, 9695, 141280, 2447592, 49212093, 1125217654, 28823053258, 817378772782, 25417591386199, 859893804621774, 31439503146486552, 1235301513403984512, 51906185332282554369, 2322562816163062723410, 110253678955655801174716, 5534198888175777261628156

Offset: 1

Peter Luschny, Aug 12 2012

nonn

See A215474 for the definitions.

			a(3) = 14 = card{000, 001, 002, 010, 011, 012, 020, 021, 022, 111, 112, 121, 122, 222}.

Alois P. Heinz, Table of n, a(n) for n = 1..387

Cf. A215474, A143328.

a[n_] := Sum[1/j Sum[MoebiusMu[j/d] n^d, {d, Divisors[j]}], {j, n}];
Array[a, 20] (* Jean-François Alcover, Jun 27 2019 *)

a(n) = sum(j=1, n, sumdiv(j, d, moebius(j/d)*n^d)/j); \\ Michel Marcus, Jun 27 2019

def A215475(n):
     return add((1/j)*add(moebius(j/d)*n^d for d in divisors(j)) for j in (1..n))
[A215475(n) for n in (1..20)]

a(n) = Sum_{j=1..n} (1/j)*Sum_{d|j} mu(j/d)*n^d.

a(n) = A215474(n,n) = A143328(n,n).

cp's OEIS Frontend

A060223 Number of orbits of length n under the map whose periodic points are counted by A000670.

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A074650 Table T(n,k) read by downward antidiagonals: number of Lyndon words (aperiodic necklaces) with n beads of k colors, n >= 1, k >= 1.

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Magma

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A281013 Tetrangle T(n,k,i) = i-th part of k-th prime composition of n.

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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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Julia

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A277427 Prime permutations, ordered lexicographically.

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A215475 Number of n-ary n-tuples (a_1,...,a_n) such that the string a_1...a_n is preprime.

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