A285548 -id:A285548 - cp's OEIS frontend

A002729 Number of equivalence classes of binary sequences of period n.

1, 2, 3, 4, 6, 6, 13, 10, 24, 22, 45, 30, 158, 74, 245, 368, 693, 522, 2637, 1610, 7386, 8868, 19401, 16770, 94484, 67562, 216275, 277534, 815558, 662370, 4500267, 2311470, 8466189, 13045108, 31593285, 40937606, 159772176, 103197490, 401913697

Offset: 0

N. J. A. Sloane

From Pab Ter (pabrlos2(AT)yahoo.com), Jan 24 2006: (Start)
The number of equivalence classes of sequences of period p, taking values in a set with b elements, is given by:
N(p) = (1/(p*phi(p)))*Sum_{t=0..p-1} Sum_{k=1..p-1 & gcd(p,k)=1} b^C(k,t) where C(k,t), the number of disjoint cycles of the permutations considered, is C(k,t) = Sum_{u=0..p-1} 1/M(k,p/gcd(p,u(k-1)+t)).
If gcd(k,L)=1, M(k,L) denotes the least positive integer M such that 1+k+...+k^(M-1) == 0 (mod L). Also if gcd(k,L)=1 and Ek(L) denotes the exponent of k mod L: M(k,L)=L*Ek(L)/gcd(L,1+k+...+k^(Ek(L)-1)).
(End)
Number of two-colored necklaces of length n, where similar necklaces are counted only once. Two necklaces of length n, given by color functions c and d from {0, ..., n-1} to N (set of natural numbers) are considered similar iff there is a factor f, 0 < f < n, satisfying gcd(f,n) = 1, such that, for all k from {0, ..., n-1}, d(f * k mod n) = c(k). I.e., the bead at position k is moved to f * k mod n. In other words: the sequence counts the orbits of the action of the multiplicative group {f | 0 < f < n, gcd(f,n) = 1} on the set of two-colored necklaces where f maps c to d with the formula above. - Matthias Engelhardt
Counts the same necklaces as A000029 but some of the necklaces viewed as distinct in A000029 are now viewed as equal. In particular, this implies that a(n) <= A000029(n) for every n.

D. Z. Dokovic, I. Kotsireas, D. Recoskie, J. Sawada, Charm bracelets and their application to the construction of periodic Golay pairs, Discrete Applied Mathematics, Volume 188, 19 June 2015, Pages 32-40.
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).

Vincenzo Librandi, Table of n, a(n) for n = 0..50
CombOS - Combinatorial Object Server, Generate necklaces, Lyndon words, chord diagrams, and relatives.
D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
M. Engelhardt, The N queens problem.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
Index entries for sequences related to necklaces

Row 2 of A285548.

Cf. A002730.

with(numtheory): M:=proc(k,L) local e,s: s:=1: for e from 1 do if(s mod L = 0) then RETURN(e) else s:=s+k^e fi od: end; C:=proc(k,t,p) local u: RETURN(add(M(k,p/igcd(p,u*(k-1)+t))^(-1),u=0..p-1)) :end; N:=proc(p) options remember: local s,t,k: if(p=1) then RETURN(2) fi: s:=0: for t from 0 to p-1 do for k from 1 to p-1 do if igcd(p,k)=1 then s:=s+2^C(k,t,p) fi od od: RETURN(s/(p*phi(p))):end;seq(N(p),p=1..51); # first M expression
with(numtheory): E:=proc(k,L) if(L=1) then RETURN(1) else RETURN(order(k,L)) fi end; M:=proc(k,L) local s,EkL: EkL:=E(k,L): if(k>1) then s:=(k^EkL-1)/(k-1): RETURN(L*EkL/igcd(L,s)) else RETURN(L*EkL/igcd(L,EkL)) fi end; C:=proc(k,t,p) local u: RETURN(add(M(k,p/igcd(p,u*(k-1)+t))^(-1),u=0..p-1)) :end; N:=proc(p) options remember: local s,t,k: if(p=1) then RETURN(2) fi: s:=0: for t from 0 to p-1 do for k from 1 to p-1 do if igcd(p,k)=1 then s:=s+2^C(k,t,p) fi od od: RETURN(s/(p*phi(p))):end;seq(N(p),p=1..51);# second M expression (Pab Ter)

max = 38; m[k_, n_] := (s = 1; Do[ If[ Mod[s, n] == 0, Return[e], s = s + k^e ] , {e, 1, max}]); c[k_, t_, n_] := Sum[ m[k, n/GCD[n, u*(k-1) + t]]^(-1), {u, 0, n-1}]; a[n_] := (s = 0; Do[ If[ GCD[n, k] == 1 , s = s + 2^c[k, t, n]] , {k, 1, n-1}, {t, 0, n-1}]; s/(n*EulerPhi[n])); a[0] = 1; a[1] = 2; Table[a[n], {n, 0, max}] (* Jean-François Alcover, Dec 06 2011, after Maple *)

Reference gives formula.

More terms from Pab Ter (pabrlos2(AT)yahoo.com), Oct 22 2005

Entry revised by N. J. A. Sloane at the suggestion of Max Alekseyev, Jun 20 2007

A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

Original entry on oeis.org

1, 1, 2, 1, 4, 3, 1, 6, 9, 4, 1, 12, 18, 16, 5, 1, 12, 54, 40, 25, 6, 1, 40, 72, 160, 75, 36, 7, 1, 28, 405, 280, 375, 126, 49, 8, 1, 96, 390, 2176, 825, 756, 196, 64, 9, 1, 104, 1944, 2800, 8125, 2016, 1372, 288, 81, 10, 1, 280, 3411, 17920, 13175, 23976, 4312, 2304, 405

Offset: 1

N. J. A. Sloane, Dec 01 2007, based on email from Max Alekseyev, Nov 08 2007

From Andrew Howroyd, Apr 22 2017: (Start)
Number of step shifted (decimated) sequences of length n using a maximum of m different symbols. See A056371 for an explanation of step shifts. -
Number of mappings with domain {0..n-1} and codomain {1..m} up to equivalence.  Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {0..n-1}. (End)

			Array begins:
1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, ...
2, 4, 6, 12, 12, 40, 28, 96, 104, 280, 216, 1248, 704, 2800, 4344, 8928, 8232, 44224, 29204, 136032, ...
3, 9, 18, 54, 72, 405, 390, 1944, 3411, 14985, 17802, 139968, 133104, 798525, 1804518, 5454378, 8072532, 64599849, 64573626, 437732424, ...
4, 16, 40, 160, 280, 2176, 2800, 17920, 44224, 263296, 419872, 4280320, 5594000, 44751616, 134391040, 539054080, 1073758360, 11453771776, 15271054960, 137575813120, ...
5, 25, 75, 375, 825, 8125, 13175, 103125, 327125, 2445625, 4884435, 61640625, 101732425, 1017323125, 3816215625, 19104609375, 47683838325, 635787765625, 1059638680675, 11924780390625, ...

Vincenzo Librandi, Table of n, a(n) for n = 1..1830
Max Alekseyev, First 20 rows and columns of table
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row m=2 is A056371
Row m=3 is A056372
Row m=4 is A056373
Row m=5 is A056374
Row m=6 is A056375
Column n=2 is A000290
Column n=3 is A002411
Column n=4 is A019582
Cf. A285522, A285548.

a[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n]==1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #]&], 0], {k, 1, n}]; Table[a[m-n+1, n], {m, 1, 15}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Dec 01 2015 *)

for(i=1,15,for(m=1,i,n=i-m+1; print1(sum(k=1, n, if(gcd(k,n)==1, m^sumdiv(n,d,eulerphi(d)/znorder(Mod(k,d))),0))/eulerphi(n)","))) \\ Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

More terms from Herman Jamke (hermanjamke(AT)fastmail.fm), Apr 26 2008

Offset corrected by Andrew Howroyd, Apr 20 2017

A288620 Triangle read by rows: T(n,k) = number of step shifted (decimated) sequence structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 5, 4, 1, 1, 5, 8, 3, 1, 1, 19, 50, 37, 9, 1, 1, 13, 54, 63, 26, 4, 1, 1, 47, 284, 479, 299, 83, 11, 1, 1, 51, 525, 1316, 1183, 454, 82, 8, 1, 1, 139, 2370, 8597, 10701, 5761, 1492, 196, 13, 1, 1, 107, 2872, 14619, 24736, 17998, 6429, 1198, 119, 6, 1

Offset: 1

Andrew Howroyd, Jun 11 2017

See A056371 for an explanation of step shifts. Permuting the symbols will not change the structure.

			Triangle begins
1;
1,  1;
1,  2,   1;
1,  5,   4,    1;
1,  5,   8,    3,    1;
1, 19,  50,   37,    9,   1;
1, 13,  54,   63,   26,   4,  1;
1, 47, 284,  479,  299,  83, 11, 1;
1, 51, 525, 1316, 1183, 454, 82, 8, 1;
...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A056396, A056397, A056398, A056399, A056400.
Row sums are A288621.
Partial row sums include A056391, A056392, A056393, A056394, A056395.
Cf. A056371, A288627, A285522, A285548, A132191.

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(StepShiftPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

A288627 Triangle read by rows: T(n,k) = number of step cyclic shifted sequence structures of length n using exactly k different symbols.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 3, 2, 1, 1, 2, 3, 1, 1, 1, 7, 14, 11, 3, 1, 1, 4, 11, 13, 6, 1, 1, 1, 13, 52, 83, 52, 18, 3, 1, 1, 10, 72, 162, 148, 59, 13, 2, 1, 1, 25, 274, 930, 1140, 630, 171, 28, 3, 1, 1, 14, 281, 1369, 2306, 1681, 612, 118, 14, 1, 1

Offset: 1

Andrew Howroyd, Jun 11 2017

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent. Permuting the symbols will not change the structure.

			Triangle begins
1;
1,  1;
1,  1,   1;
1,  3,   2,   1;
1,  2,   3,   1,    1;
1,  7,  14,  11,    3,   1;
1,  4,  11,  13,    6,   1,   1;
1, 13,  52,  83,   52,  18,   3,  1;
1, 10,  72, 162,  148,  59,  13,  2, 1;
1, 25, 274, 930, 1140, 630, 171, 28, 3, 1;
...

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Columns 2-6 are A056434, A056435, A056436, A056437, A056438.
Row sums are A288628.
Partial row sums include A056429, A056430, A056431, A056432, A056433.
Cf. A056391, A056371, A288620, A285522, A285548, A132191.

\\ see A056391 for Polya enumeration functions
T(n,k) = NonequivalentStructsExactly(CyclicStepShiftPerms(n), k); \\ Andrew Howroyd, Oct 14 2017

A056411 Number of step cyclic shifted sequences using a maximum of three different symbols.

Original entry on oeis.org

3, 6, 10, 21, 24, 92, 78, 327, 443, 1632, 1698, 12769, 10464, 57840, 122822, 348222, 476052, 3597442, 3401970, 22006959, 41597374, 142677588, 186077886, 1476697627, 1694658003, 8147282460, 15690973754, 68149816689, 84520682160, 857935531804, 664166389302, 3620293575942, 8422974597554, 30656600391720, 59561470990362

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 3 of A285548.
Cf. A002729.
Cf. A056412, A056413, A056414.

M[j_, L_] := Module[{m=1}, While[Sum[j^i, {i, 0, m-1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[1/M[j, n/GCD[n, u*(j-1)+t]], {u, 0, n-1}]; CB[n_, k_] = If [n==1, k, 1/(n*EulerPhi[n])*Sum[If[1==GCD[n, j], k^c[j, t, n], 0], {t, 0, n-1}, {j, 1, n-1}]]; Table[Print[cb = CB[n, 3]]; cb, {n, 1, 35}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,3),", "));
\\ second argument k=3, 4, 5, 6 respectively gives A056411, A056412, A056413, A056414.
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A056412 Number of step cyclic shifted sequences using a maximum of four different symbols.

Original entry on oeis.org

4, 10, 20, 55, 76, 430, 460, 2605, 5164, 26962, 38572, 367645, 431780, 3203430, 8993804, 33860125, 63177820, 636462350, 803796700, 6886280971, 17456594380, 79965550558, 139069427020, 1466861706095, 2251803181492, 14434628481170, 37066691779180, 214483458079665, 354963555781060, 4803855154772166

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 4 of A285548.

Cf. A002729.

M[j_, L_] := Module[{m = 1}, While[Sum[j^i, {i, 0, m-1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[1/M[j, n / GCD[n, u*(j-1) + t]], {u, 0, n - 1}]; CB[n_, k_] = If[n==1, k, 1/(n*EulerPhi[n]) * Sum[ If[1 == GCD[n, j], k^c[j, t, n], 0] , {t, 0, n-1}, {j, 1, n-1}]]; Table[Print[cb = CB[n, 4]]; cb, {n, 1, 30}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,4),", "));
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A056413 Number of step cyclic shifted sequences using a maximum of five different symbols.

Original entry on oeis.org

5, 15, 35, 120, 201, 1505, 2015, 14070, 37085, 246753, 445515, 5205790, 7832185, 72703645, 254689657, 1196213445, 2805046965, 35322811755, 55770979195, 596439735024, 1892294578755, 10837223014665, 23559159229935, 310484619147940, 596046508875701, 4776013513099405, 15330413466776835, 110874578286500410

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 5 of A285548.

Cf. A002729.

M[j_, L_] := Module[{m = 1}, While[Sum[ j^i, {i, 0, m - 1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[ 1/M[j, n / GCD[n, u*(j - 1) + t] ], {u, 0, n - 1} ]; CB[n_, k_] = If [n == 1, k, 1/(n*EulerPhi[n]) * Sum[ If[1 == GCD[n, j], k^c[j, t, n], 0] , {t, 0, n-1}, {j, 1, n-1}]]; Table[ Print[ cb = CB[n, 5]]; cb, {n, 1, 28}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,5),", "));
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A056414 Number of step cyclic shifted sequences using a maximum of six different symbols.

Original entry on oeis.org

6, 21, 56, 231, 462, 4291, 6966, 57561, 188866, 1519035, 3302922, 45921281, 83747286, 933081411, 3920355712, 22075451286, 62230996506, 940379310731, 1781757016326, 22856965214727, 87052415641136, 598280600648031, 1560731765058606, 24680195365765751, 56860576713326910, 546736312124316741, 2105947271634851386

Offset: 1

Marks R. Nester

nonn

See A056371 for an explanation of step shifts. Under step cyclic shifts, abcde, bdace, bcdea, cdeab and daceb etc. are equivalent.

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

D. Z. Dokovic, I. Kotsireas et al., Charm bracelets and their application to the construction of periodic Golay pairs, arXiv:1405.7328 [math.CO], 2014.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.

Row 6 of A285548.

Cf. A002729.

M[j_, L_] := Module[{m = 1}, While[Sum[ j^i, {i, 0, m - 1}] ~Mod~ L != 0, m++]; m]; c[j_, t_, n_] := Sum[ 1/M[j, n / GCD[n, u*(j - 1) + t] ], {u, 0, n - 1}]; CB[n_, k_] = If[n == 1, k, 1/(n*EulerPhi[n]) * Sum[ If[1 == GCD[n, j], k^c[j, t, n], 0], {t, 0, n-1}, {j, 1, n-1}]]; Table[ Print[ cb = CB[n, 6]]; cb, {n, 1, 27}] (* Jean-François Alcover, Dec 04 2015, after Joerg Arndt *)

\\ see p.3 of the Dokovic et al. reference
M(j,  L)={my(m=1); while ( sum(i=0, m-1, j^i) % L != 0, m+=1 ); m; }
c(j, t, n)=sum(u=0,n-1, 1/M(j, n / gcd(n, u*(j-1)+t) ) );
CB(n, k)=if (n==1,k, 1/(n*eulerphi(n)) * sum(t=0,n-1, sum(j=1,n-1, if(1==gcd(n,j), k^c(j,t,n), 0) ) ) );
for(n=1, 66, print1(CB(n,6),", "));
\\ Joerg Arndt, Aug 27 2014

Refer to Titsworth or slight "simplification" in Nester.

Added more terms, Joerg Arndt, Aug 27 2014

A285522 Array read by antidiagonals: T(m,n) = number of circulant digraphs up to Cayley isomorphism on n vertices with edges colored according to step value using a maximum of m-1 colors.

Original entry on oeis.org

1, 1, 1, 1, 2, 1, 1, 3, 3, 1, 1, 6, 6, 4, 1, 1, 6, 18, 10, 5, 1, 1, 20, 24, 40, 15, 6, 1, 1, 14, 135, 70, 75, 21, 7, 1, 1, 48, 130, 544, 165, 126, 28, 8, 1, 1, 52, 648, 700, 1625, 336, 196, 36, 9, 1, 1, 140, 1137, 4480, 2635, 3996, 616, 288, 45, 10, 1

Offset: 1

Andrew Howroyd, Apr 20 2017

For the base case of m=2 the sequence counts circulant digraphs up to Cayley isomorphism. Two circulant graphs are Cayley isomorphic if there is a d, which is necessarily prime to n, that transforms through multiplication modulo n the step values of one graph into those of the other. For squarefree n this is the only way that two circulant graphs can be isomorphic. (See Liskovets reference for a proof.)
Alternatively, the number of mappings with domain {1..n-1} and codomain {1..m} up to equivalence. Mappings A and B are equivalent if there is a d, prime to n, such that A(i) = B(i*d mod n) for i in {1..n-1}. This sequence differs from A132191 only in that sequence also includes 0 in the domain which introduces an extra factor of m into the results since zero multiplied by anything is zero.
All column sequences are polynomials of order n-1 and these are the cycle index polynomials.
This sequence is also related to A075195(n, m) which counts necklaces and A285548(m, n) which is the sequence described in the Titsworth reference. In particular, A075195 is the analogous array with equivalence determined through the additive group instead of by multiplication whereas A285548 allows for both addition and multiplication.

			Table starts:
\n  1 2  3   4    5    6    7     8      9      10
m\ ---------------------------------------------------
1 | 1 1  1   1    1    1    1     1      1       1 ...
2 | 1 2  3   6    6   20   14    48     52     140 ...
3 | 1 3  6  18   24  135  130   648   1137    4995 ...
4 | 1 4 10  40   70  544  700  4480  11056   65824 ...
5 | 1 5 15  75  165 1625 2635 20625  65425  489125 ...
6 | 1 6 21 126  336 3996 7826 72576 280596 2521476 ...
...
Case n=10:
Only 1, 3, 7, 9 are prime to 10.
Multiplication modulo 10 is described by the following multiplication table.
  1, 2, 3, 4, 5, 6, 7, 8, 9  => (1)(2)(3)(4)(5)(6)(7)(8)(9) => m^9
  3, 6, 9, 2, 5, 8, 1, 4, 7  => (1397)(2684)(5)             => m^3
  7, 4, 1, 8, 5, 2, 9, 6, 3  => (1793)(2486)(5)             => m^3
  9, 8, 7, 6, 5, 4, 3, 2, 1  => (19)(28)(37)(46)(5)         => m^5
Each row of the multiplication table can be viewed as a permutation and together these form a commutative group on 4 elements. In this case the group is isomorphic to the cyclic group C_4. Each permutation can be represented in cycle notation. (shown above to the right of the corresponding multiplication table row). In order to count the equivalence classes using Polya's enumeration theorem only the number of cycles in each permutation is needed.
This gives the cycle index polynomial (1/4)*(m^9 + m^5 + 2*m^3). Putting m = 1..4 gives 1, 140, 4995, 65824.

Andrew Howroyd, Table of n, a(n) for n = 1..1275
V. A. Liskovets and R. Poeschel, On the enumeration of circulant graphs of prime-power and squarefree orders.
R. C. Titsworth, Equivalence classes of periodic sequences, Illinois J. Math., 8 (1964), 266-270.
Eric Weisstein's World of Mathematics, Circulant Graph.
Wikipedia, Polya enumeration theorem.

Row 2 is A056391.
Columns include A002411, A006528, A168178, A006565, A282613, A054624.
Cf. A132191, A075195, A285548, A002729.

A132191[m_, n_] := (1/EulerPhi[n])*Sum[If[GCD[k, n] == 1, m^DivisorSum[n, EulerPhi[#] / MultiplicativeOrder[k, #] &], 0], {k, 1, n}];
T[m_, n_] := A132191[m, n]/m;
Table[T[m - n + 1, n], {m, 1, 11}, {n, m, 1, -1}] // Flatten (* Jean-François Alcover, Jun 06 2017 *)

a(n,x)=sum(k=1, n, if(gcd(k, n)==1, x^(sumdiv(n, d, eulerphi(d)/znorder(Mod(k, d)))-1), 0))/eulerphi(n);
for(m=1, 6, for(n=1, 10, print1( a(n,m), ", ") ); print(); );

T(m, n) = A132191(m, n) / m.

cp's OEIS Frontend

A002729 Number of equivalence classes of binary sequences of period n.

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A132191 Square array a(m,n) read by antidiagonals, defined by A000010(n)*a(m,n) = Sum_{k=1..n, gcd(k,n)=1} m^{ Sum_{d|n} A000010(d)/ (multiplicative order of k modulo d) }.

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A288620 Triangle read by rows: T(n,k) = number of step shifted (decimated) sequence structures of length n using exactly k different symbols.

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A288627 Triangle read by rows: T(n,k) = number of step cyclic shifted sequence structures of length n using exactly k different symbols.

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A056411 Number of step cyclic shifted sequences using a maximum of three different symbols.

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A056412 Number of step cyclic shifted sequences using a maximum of four different symbols.

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A056413 Number of step cyclic shifted sequences using a maximum of five different symbols.

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