A074650 -id:A074650 - cp's OEIS frontend

A320075 Number of length n primitive (=aperiodic or period n) 10-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

1, 9, 99, 990, 9999, 99891, 999999, 9999000, 99999900, 999989991, 9999999999, 99999899010, 999999999999, 9999998999991, 99999999989901, 999999990000000, 9999999999999999, 99999999899900100, 999999999999999999, 9999999998999999010, 99999999999998999901

Offset: 1

Alois P. Heinz, Oct 05 2018

nonn

Dirichlet convolution of mu(n) with 10^(n-1).

Alois P. Heinz, Table of n, a(n) for n = 1..1001

Column k=10 of A143325.
First differences of A320094.
Cf. A008683, A074650, A143324.

a:= n-> add(`if`(d=n, 10^(n-1), -a(d)), d=numtheory[divisors](n)):
seq(a(n), n=1..25);

a(n) = Sum_{d|n} 10^(d-1) * mu(n/d).

a(n) = 10^(n-1) - Sum_{d

a(n) = A143325(n,10).

a(n) = A074650(n,10) * n/10.

a(n) = A143324(n,10) / 10.

G.f.: Sum_{k>=1} mu(k)*x^k/(1 - 10*x^k). - Ilya Gutkovskiy, Oct 25 2018

A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

Original entry on oeis.org

1, 2, -1, 3, -3, 0, 4, -6, 2, 0, 5, -10, 8, -3, 0, 6, -15, 20, -18, 6, 0, 7, -21, 40, -60, 48, -11, 0, 8, -28, 70, -150, 204, -124, 18, 0, 9, -36, 112, -315, 624, -690, 312, -30, 0, 10, -45, 168, -588, 1554, -2620, 2340, -810, 56, 0, 11, -55, 240, -1008, 3360, -7805, 11160, -8160, 2184, -105, 0

Offset: 1

Seiichi Manyama, Apr 12 2025

			Square array begins:
   1,   2,    3,    4,     5,     6,      7, ...
  -1,  -3,   -6,  -10,   -15,   -21,    -28, ...
   0,   2,    8,   20,    40,    70,    112, ...
   0,  -3,  -18,  -60,  -150,  -315,   -588, ...
   0,   6,   48,  204,   624,  1554,   3360, ...
   0, -11, -124, -690, -2620, -7805, -19656, ...
   0,  18,  312, 2340, 11160, 39990, 117648, ...

Columns k=1..5 give A154955, A038063, A038064, A038065, A038066.
Main diagonal gives A383012.
Cf. A008683, A074650.

a(n, k) = -sumdiv(n, d, moebius(n/d)*(-k)^d)/n;

G.f. of column k: Sum_{j>=1} mu(j) * log(1 + k*x^j) / j.

Product_{n>=1} 1/(1 - x^n)^A(n,k) = 1 + k*x.

A060216 Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022).

Original entry on oeis.org

13, 78, 728, 7098, 74256, 804076, 8964072, 101962770, 1178277464, 13785812040, 162923672184, 1941506688940, 23298085122480, 281241165925044, 3412392867581152, 41588538022965570

Offset: 1

Thomas Ward, Mar 21 2001

nonn

Number of monic irreducible polynomials of degree n over GF(13). - Robert Israel, Jan 07 2015

Number of Lyndon words (aperiodic necklaces) with n beads of 13 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=78 since there are 169 points of period 2 in the full 13-shift and 13 fixed points, so there must be (169-13)/2 = 78 orbits of length 2.

Indranil Ghosh, Table of n, a(n) for n = 1..100
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

Column 13 of A074650.

Cf. A001022.

f:= n -> add(numtheory:-mobius(d)*13^(n/d),d=numtheory:-divisors(n))/n;
seq(f(n), n=1..100); # Robert Israel, Jan 07 2015

a[n_]:=(1/n) * Sum[MoebiusMu[d] *13^(n/d), {d, Divisors[n]}]; Table[a[n], {n, 20}] (* Indranil Ghosh, Mar 26 2017 *)

a(n) = sumdiv(n, d, moebius(d)*13^(n/d))/n; \\ Michel Marcus, Jan 07 2015

from sympy import divisors, mobius
print([sum(mobius(d) * 13**(n//d) for d in divisors(n))//n for n in range(1, 21)]) # Indranil Ghosh, Mar 26 2017

a(n) = (1/n)* Sum_{d|n} mu(d) 13^(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 13*x^k))/k. - Ilya Gutkovskiy, May 19 2019

A060217 Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023).

Original entry on oeis.org

14, 91, 910, 9555, 107562, 1254435, 15059070, 184468830, 2295671560, 28925411697, 368142288150, 4724492067295, 61054982558010, 793714765724595, 10371206370484778, 136122083520848880, 1793608631137129170, 23715491899442676060, 314542313628890231430, 4183412771249777343369

Offset: 1

Thomas Ward, Mar 21 2001

Number of Lyndon words (aperiodic necklaces) with n beads of 14 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=91 since there are 196 points of period 2 in the full 14-shift and 14 fixed points, so there must be (196-14)/2 = 91 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..870
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 14 of A074650.

Cf. A001023.

A060217:= func< n | (&+[MoebiusMu(d)*14^Floor(n/d): d in Divisors(n)])/n >;
[A060217(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024

A060217[n_]:= DivisorSum[n, MoebiusMu[#]*14^(n/#) &]/n;
Table[A060217[n], {n,40}] (* G. C. Greubel, Aug 01 2024 *)

a001023(n) = 14^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001023(n/d)); \\ Michel Marcus, Sep 11 2017

def A060217(n): return sum(moebius(k)*14^(n//k) for k in (1..n) if (k).divides(n))/n
[A060217(n) for n in range(1,41)] # G. C. Greubel, Aug 01 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001023(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 14*x^k))/k. - Ilya Gutkovskiy, May 19 2019

More terms from Michel Marcus, Sep 11 2017

A060218 Number of orbits of length n under the full 15-shift (whose periodic points are counted by A001024).

Original entry on oeis.org

15, 105, 1120, 12600, 151872, 1897840, 24408480, 320355000, 4271484000, 57664963104, 786341441760, 10812193870800, 149707312950720, 2085208989609360, 29192926025339776, 410525522071875000, 5795654431511374080, 82105104444274758000, 1166756747396368729440, 16626283650369421872480

Offset: 1

Thomas Ward, Mar 21 2001

Number of Lyndon words (aperiodic necklaces) with n beads of 15 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=105 since there are 225 points of period 2 in the full 15-shift and 15 fixed points, so there must be (225-15)/2 = 105 orbits of length 2.

Robert Israel, Table of n, a(n) for n = 1..851
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 15 of A074650.

Cf. A001024.

A060218:= func< n | (&+[MoebiusMu(d)*15^Floor(n/d): d in Divisors(n)])/n >;
[A060218(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024

f:= n -> 1/n*add(numtheory:-mobius(d)*15^(n/d), d = numtheory:-divisors(n)):
map(f, [$1..30]); # Robert Israel, Oct 28 2018

A060218[n_]:= DivisorSum[n, MoebiusMu[#]*15^(n/#) &]/n;
Table[A060218[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)

a001024(n) = 15^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017

def A060218(n): return sum(moebius(k)*15^(n//k) for k in (1..n) if (k).divides(n))/n
[A060218(n) for n in range(1,41)] # G. C. Greubel, Aug 01 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001024(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 15*x^k))/k. - Ilya Gutkovskiy, May 19 2019

More terms from Michel Marcus, Sep 11 2017

A060219 Number of orbits of length n under the full 16-shift (whose periodic points are counted by A001025).

Original entry on oeis.org

16, 120, 1360, 16320, 209712, 2795480, 38347920, 536862720, 7635496960, 109951057896, 1599289640400, 23456246655680, 346430740566960, 5146970983535160, 76861433640386288, 1152921504338411520, 17361641481138401520, 262353693488939386880, 3976729669784964390480

Offset: 1

Thomas Ward, Mar 21 2001

Number of monic irreducible polynomials of degree n over GF(16). - Robert Israel, Jan 07 2015

Number of Lyndon words (aperiodic necklaces) with n beads of 16 colors. - Andrew Howroyd, Dec 10 2017

			a(2)=120 since there are 256 points of period 2 in the full 16-shift and 16 fixed points, so there must be (256-16)/2 = 120 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..825
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

Column 16 of A074650.

Cf. A001025.

A060219:= func< n | (&+[MoebiusMu(d)*16^Floor(n/d): d in Divisors(n)])/n >;
[A060219(n): n in [1..40]]; // G. C. Greubel, Aug 01 2024

f:= (n,p) -> add(numtheory:-mobius(d)*p^(n/d),d=numtheory:-divisors(n))/n:
seq(f(n,16),n=1..30); # Robert Israel, Jan 07 2015

A060219[n_]:= DivisorSum[n, MoebiusMu[#]*16^(n/#) &]/n;Table[A060219[n], {n, 40}] (* G. C. Greubel, Aug 01 2024 *)

a(n) = sumdiv(n, d, moebius(d)*16^(n/d))/n; \\ Michel Marcus, Jan 07 2015

def A060219(n): return sum(moebius(k)*16^(n//k) for k in (1..n) if (k).divides(n))/n
[A060219(n) for n in range(1, 41)] # G. C. Greubel, Aug 01 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*16^(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 16*x^k))/k. - Ilya Gutkovskiy, May 19 2019

Terms a(17) onward added by G. C. Greubel, Aug 01 2024

A060220 Number of orbits of length n under the full 17-shift (whose periodic points are counted by A001026).

Original entry on oeis.org

17, 136, 1632, 20808, 283968, 4022064, 58619808, 871959240, 13176430176, 201599248032, 3115626937056, 48551851084080, 761890617915840, 12026987582075856, 190828203433892736, 3041324491793194440, 48661191875666868480, 781282469552728498992, 12582759772902701307744

Offset: 1

Thomas Ward, Mar 21 2001

Number of monic irreducible polynomials of degree n over GF(17). - Andrew Howroyd, Dec 10 2017

			a(2)=136 since there are 289 points of period 2 in the full 17-shift and 17 fixed points, so there must be (289-17)/2 = 136 orbits of length 2.

G. C. Greubel, Table of n, a(n) for n = 1..810
Y. Puri and T. Ward, Arithmetic and growth of periodic orbits, J. Integer Seqs., Vol. 4 (2001), #01.2.1.
Yash Puri and Thomas Ward, A dynamical property unique to the Lucas sequence, Fibonacci Quarterly, Volume 39, Number 5 (November 2001), pp. 398-402.
T. Ward, Exactly realizable sequences

Column 17 of A074650.

Cf. A001026, A008683.

A060220:= func< n | (1/n)*(&+[MoebiusMu(d)*(17)^Floor(n/d): d in Divisors(n)]) >;
[A060220(n): n in [1..40]]; // G. C. Greubel, Sep 13 2024

A060220[n_]:= DivisorSum[n, (17)^(n/#)*MoebiusMu[#] &]/n;
Table[A060220[n], {n,40}] (* G. C. Greubel, Sep 13 2024 *)

a001024(n) = 17^n;
a(n) = (1/n)*sumdiv(n, d, moebius(d)*a001024(n/d)); \\ Michel Marcus, Sep 11 2017

def A060220(n): return (1/n)*sum(moebius(k)*(17)^(n/k) for k in (1..n) if (k).divides(n))
[A060220(n) for n in range(1,41)] # G. C. Greubel, Sep 13 2024

a(n) = (1/n)* Sum_{d|n} mu(d)*A001026(n/d).

G.f.: Sum_{k>=1} mu(k)*log(1/(1 - 17*x^k))/k. - Ilya Gutkovskiy, May 20 2019

More terms from Michel Marcus, Sep 11 2017

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A032165 Number of aperiodic necklaces of n beads of 10 colors.

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A032166 Number of aperiodic necklaces of n beads of 11 colors.

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A032167 Number of aperiodic necklaces of n beads of 12 colors.

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A320075 Number of length n primitive (=aperiodic or period n) 10-ary words which are earlier in lexicographic order than any other word derived by cyclic shifts of the alphabet.

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A383011 Square array A(n,k), n >= 1, k >= 1, read by antidiagonals downwards, where A(n,k) = -(1/n) * Sum_{d|n} mu(n/d) * (-k)^d.

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A060216 Number of orbits of length n under the full 13-shift (whose periodic points are counted by A001022).

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A060217 Number of orbits of length n under the full 14-shift (whose periodic points are counted by A001023).

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