A102190 -id:A102190 - cp's OEIS frontend

A032193 Number of necklaces with 8 black beads and n-8 white beads.

1, 1, 5, 15, 43, 99, 217, 429, 810, 1430, 2438, 3978, 6310, 9690, 14550, 21318, 30667, 43263, 60115, 82225, 111041, 148005, 195143, 254475, 328756, 420732, 534076, 672452, 840652, 1043460, 1287036, 1577532, 1922741

Offset: 8

Christian G. Bower

nonn

The g.f. is Z(C_8,x)/x^8, the 8-variate cycle index polynomial for the cyclic group C_8, with substitution x[i]->1/(1-x^i), i=1,...,8. Therefore by Polya enumeration a(n+8) is the number of cyclically inequivalent 8-necklaces whose 8 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_8,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005
From Petros Hadjicostas, Aug 31 2018: (Start)
The CIK[k] transform of sequence (c(n): n>=1) has generating function A_k(x) = (1/k)*Sum_{d|k} phi(d)*C(x^d)^{k/d}, where C(x) = Sum_{n>=1} c(n)*x^n is the g.f. of (c(n): n>=1).
When c(n) = 1 for all n >= 1, we get C(x) = x/(1-x) and A_k(x) = (x^k/k)*Sum_{d|k} phi(d)*(1-x^d)^{-k/d}, which is the g.f. of the number a_k(n) of necklaces of n beads of 2 colors with k of them black and n-k of them white.
Using Taylor expansions, we can easily prove that a_k(n) = (1/k)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d - 1, k/d - 1) = (1/n)*Sum_{d|gcd(n,k)} phi(d)*binomial(n/d, k/d), which is Robert A. Russell's formula in the Mathematica code below.
For this sequence k = 8, and thus we get the formulae below.
(End)

Christian G. Bower, Transforms (2)
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-4,-4,11,-8,0,8,-10,0,8,0,-10,8,0,-8,11,-4,-4,4,-1).

Column k=8 of A047996.

Cf. A004526, A005514, A007997, A008610, A008646, A032191, A032192.

k = 8; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)
CoefficientList[Series[1/8*(1/(1 - x)^8 + 1/(1 - x^2)^4 + 2/(1 - x^4)^2 + 4/(1 - x^8)^1),{x, 0, 30}], x] (* Stefano Spezia, Sep 01 2018 *)

"CIK[ 8 ]" (necklace, indistinct, unlabeled, 8 parts) transform of 1, 1, 1, 1...
G.f.: (x^8)*(1-3*x+5*x^2+3*x^3-4*x^4+4*x^5+6*x^6-4*x^7+7*x^8-x^9+x^10+x^11)/((1-x)^4*(1-x^2)^2*(1-x^4)*(1-x^8)).
G.f.: 1/8*x^8*(1/(1-x)^8+1/(1-x^2)^4+2/(1-x^4)^2+4/(1-x^8)^1). - Herbert Kociemba, Oct 22 2016
a(n) = (1/8)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d - 1, 8/d - 1) = (1/n)*Sum_{d|gcd(n,8)} phi(d)*binomial(n/d, 8/d). - Petros Hadjicostas, Aug 31 2018

A328858 Numbers with a record number of distinct values of the Euler totient function applied to their divisors (A319696).

Original entry on oeis.org

1, 3, 8, 15, 32, 45, 72, 105, 144, 216, 288, 432, 792, 864, 1296, 1584, 1728, 2376, 2592, 3168, 4752, 9504, 14256, 19008, 28512, 38016, 54000, 57024, 85536, 108000, 114048, 162000, 171072, 216000, 218592, 228096, 324000, 342144, 437184, 465696, 648000, 655776

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

The corresponding record values are 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 13, 14, 16, 17, 18, 19, 20, 21, 22, 26, 32, 34, 38, 42, 44, 45, 50, 52, 54, 58, 59, 62, 63, 64, 66, 71, 72, 76, 78, 83, 84, ...

			The first 10 terms of A319696(k) are 1, 1, 2, 2, 2, 2, 2, 3, 3, 2. The record values 1, 2, and 3 are obtained at k = 1, 3, and 8. Therefore this sequence begins with 1, 3, 8.

Amiram Eldar, Table of n, a(n) for n = 1..100

Cf. A000010, A102190, A319696.

f[n_] := Length @ Union[EulerPhi /@ Divisors[n]]; fm = 0; s = {}; Do[f1 = f[n]; If[f1 > fm, fm = f1; AppendTo[s, n]], {n, 1, 660000}]; s

A032194 Number of necklaces with 9 black beads and n-9 white beads.

Original entry on oeis.org

1, 1, 5, 19, 55, 143, 335, 715, 1430, 2704, 4862, 8398, 14000, 22610, 35530, 54484, 81719, 120175, 173593, 246675, 345345, 476913, 650325, 876525, 1168710, 1542684, 2017356, 2615104, 3362260, 4289780, 5433736, 6835972

Offset: 9

Christian G. Bower

nonn

The g.f. is Z(C_9,x)/x^9, the 9-variate cycle index polynomial for the cyclic group C_9, with substitution x[i]->1/(1-x^i), i=1,...,9. Therefore by Polya enumeration a(n+9) is the number of cyclically inequivalent 9-necklaces whose 9 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_9,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

Christian G. Bower, Transforms (2)
David Broadhurst and Xavier Roulleau, Number of partitions of modular integers, arXiv:2502.19523 [math.NT], 2025. See p. 19.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
Frank Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (6,-15,22,-27,36,-42,36,-27,23,-21,21,-23,27,-36,42,-36,27,-22,15,-6,1).

Column k=9 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193.

k = 9; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 9 ]" (necklace, indistinct, unlabeled, 9 parts) transform of 1, 1, 1, 1...
G.f.: (x^9)*(1-5*x+14*x^2-18*x^3+21*x^4-21*x^5+25*x^6 -21*x^7 +21*x^8 -18*x^9 +14*x^10 -5*x^11 +x^12) / ((1-x)^6*(1-x^3)^2*(1-x^9)).
G.f.: (1/9)*x^9*(1/(1-x)^9+2/(1-x^3)^3+6/(1-x^9)^1). - Herbert Kociemba, Oct 22 2016

A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.

Original entry on oeis.org

1, 1, 1, 2, 0, 1, 2, 0, 1, 0, 1, 4, 0, 0, 0, 0, 0, 1, 2, 0, 0, 2, 0, 0, 1, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 4, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 0, 0, 0, 1, 6, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0

Offset: 1

Wolfdieter Lang, Jun 04 2012

The partitions are ordered like in Abramowitz-Stegun (for the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used).
The row lengths sequence is A000041. The number of nonzero entries in row nr. n is  A000005(n).
The cycle index (multivariate polynomial) for the cyclic group C_n, called Z(C_n), is (sum(phi(k)*x_k^{n/k} ,k divides n))/n, n>=1, with Euler's totient function phi(n)= A000010(n). See the Harary and Palmer reference. For the coefficients of Z(C_n) in different tabulations see also A054523 and A102190.

			n\k  1 2 3 4 5 6 7 8 9 10 11 ...
1:   1
2:   1 1
3:   2 0 1
4:   2 0 1 0 1
5:   4 0 0 0 0 0 1
6:   2 0 0 2 0 0 1 0 0  0  1
...
See the link for rows n=1..8 and the Z(C_n) polynomials for n=1..15.
n=6: Z(C_6) = (2*x[6] + 2*x[3]^2 + 1*x[2]^3 + x[1]^6)/6, because the relevant partitions of 6 appear for k=1: 6, k=4: 3^2, k=7: 2^3 and k=11: 1^6

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).

Wolfdieter Lang, Cycle index Z(C_n), n=1..15.

Cf. A000005, A000010, A054523, A102190.

The cycle index polynomial for the cyclic group C_n is Z(C_n) = (a(n,k)*x[1]^(e[k,1])*x[2]^(e[k,2])*...*x[n]^(e[k,n]))/n, n>=1, if the k-th partition of n in Abramowitz-Stegun order is 1^(e[k,1]) 2^(e[k,2]) ... n^(e[k,n]), where a part j with vanishing exponent e[k,j] has to be omitted. The n dependence of the exponents has been suppressed. See the comment above for the Z(C_n) formula and the link for these polynomials for n=1..15.

a(n,k) is the coefficient the term of n*Z(C_n) corresponding to the k-th partition of n in Abramowitz-Stegun order. a(n,k) = 0 if there is no such term in Z(C_n).

A328859 Indices k of records of low value in the ratios A319696(k)/A000005(k) between the number of distinct values of the Euler totient function applied to the divisors of k and the number of divisors of k.

Original entry on oeis.org

1, 2, 60, 120, 240, 480, 960, 1920, 3840, 4080, 8160, 16320, 32640, 65280, 130560, 261120, 522240, 1044480, 1485120, 2227680, 2970240, 4455360, 8910720, 17821440, 35642880, 42325920, 63488880, 69090840, 84651840, 126977760, 169303680, 253955520, 507911040, 761866560

Offset: 1

Amiram Eldar, Oct 28 2019

nonn

The maximal possible value of the ratio A319696(k)/A000005(k) is 1 which occurs at the terms of A326835.

The rounded values of the corresponding ratios are 1, 0.5, 0.417, 0.375, 0.35, 0.333, 0.321, 0.313, 0.306, 0.275, 0.25, 0.232, 0.219, 0.208, 0.2, 0.193, 0.188, 0.183, 0.179, 0.170, 0.168, 0.158, 0.148, 0.141, 0.135, 0.132, 0.130, 0.129, 0.122, 0.117, 0.115, 0.108, 0.102, 0.101, ...

Cf. A000010, A102190, A319696, A326835.

r[n_] := Length @ Union[EulerPhi /@ (d = Divisors[n])]/Length[d]; rm = 2; s = {}; Do[r1 = r[n]; If[r1 < rm, rm = r1; AppendTo[s, n]], {n, 1, 10^5}]; s

A373527 Odd numbers k such that k and k+2 both have at least two divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

2107, 11275, 42651, 68733, 90153, 99123, 123633, 213003, 226825, 242305, 262143, 272853, 292873, 295405, 308007, 313443, 376675, 376803, 378693, 390115, 427425, 471293, 473263, 524797, 525481, 556983, 579535, 591325, 618469, 638163, 663325, 669123, 699853, 731815

Offset: 1

Amiram Eldar, Jun 08 2024

nonn

Numbers k such that k and k+2 are both in A359563.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A359563.
A373528 is a subsequence.
Cf. A000010, A102190.

q[n_] := q[n] = UnsameQ @@ EulerPhi[Divisors[n]]; Select[Range[1, 10^6, 2], ! q[#] && ! q[# + 2] &]

is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2); forstep(k = 3, kmax, 2, q2 = is(k); if(q1 && q2, print1(k-2, ", ")); q1 = q2);}

A373528 Odd numbers k such that k, k+2 and k+4 all have at least two divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

4142435, 26196331, 77118741, 89690821, 102974571, 196054673, 201060275, 206568171, 277322153, 280039833, 401784953, 402492695, 415097613, 437290371, 515636303, 526721895, 534746581, 549806211, 575090395, 580329603, 625833871, 629588043, 702183625, 710983971, 716133481

Offset: 1

Amiram Eldar, Jun 08 2024

nonn

Numbers k such that k, k+2 and k+4 are all in A359563.

Amiram Eldar, Table of n, a(n) for n = 1..100

Subsequence of A359563 and A373527.

Cf. A000010, A102190.

q[n_] := !UnsameQ @@ EulerPhi[Divisors[n]]; seq[kmax_] := Module[{tri = q /@ {1, 3, 5}, s = {}, k = 7}, While[k < kmax, If[And @@ tri, AppendTo[s, k - 6]]; tri = Join[Rest[tri], {q[k]}]; k+=2]; s]; seq[3*10^7]

is(k) = k>1 && k%2 && numdiv(k) > #Set(apply(x->eulerphi(x), divisors(k)));
lista(kmax) = {my(q1 = 0, q2 = 0, q3); forstep(k = 5, kmax, 2, q3 = is(k); if(q1 && q2 && q3, print1(k-4, ", ")); q1 = q2; q2 = q3);}

A373529 Numbers k such that k and k+1 both have at least three divisors with the same value of the Euler totient function (A000010).

Original entry on oeis.org

32319, 111320, 175959, 179360, 191919, 212120, 246519, 254079, 254960, 279279, 319599, 355508, 357399, 398600, 436149, 463239, 512000, 520064, 524799, 542240, 580040, 606879, 657152, 678699, 685880, 701631, 718640, 726920, 739556, 750519, 759759, 775775, 787815

Offset: 1

Amiram Eldar, Jun 08 2024

nonn

Numbers k such that k and k+1 are both in A359565.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Subsequence of A359565.
A373530 is a subsequence.
Cf. A000010, A102190.

q[n_] := q[n] = Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]] > 2; Select[Range[3*10^6], q[#] && q[# + 1] &]

is(k) = vecmax(matreduce(apply(x->eulerphi(x), divisors(k)))[,2]) > 2;
lista(kmax) = {my(q1 = 0, q2); for(k = 2, kmax, q2 = is(k); if(q1 && q2, print1(k-1, ", ")); q1 = q2);}

A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).

Original entry on oeis.org

1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 2, 1, 2, 1, 3, 1, 2, 1, 3, 1, 2, 1

Offset: 1

Amiram Eldar, Jun 08 2024

The sums of the first 10^k terms, for k = 1, 2, ..., are 15, 161, 1641, 16554, 166029, 1662306, 16630535, 166335597, 1663473941, 16635216306, ... . Apparently, this sequence has an asymptotic mean 1.663... .

			a(2) = 2 since 2 has 2 divisors, 1 and 2, and phi(1) = phi(2) = 1.
a(12) = 3 since 3 of the divisors of 12 (3, 4 and 6) have the same value of phi: phi(3) = phi(4) = phi(6) = 2.

Amiram Eldar, Table of n, a(n) for n = 1..10000

Cf. A000010, A102190, A319696, A326835, A359563, A359565, A373532.

a[n_] := Max[Tally[EulerPhi[Divisors[n]]][[;; , 2]]]; Array[a, 100]

a(n) = vecmax(matreduce(apply(x->eulerphi(x), divisors(n)))[ , 2]);

from collections import Counter
from sympy import divisors, totient
def a(n):
    c = Counter(totient(d) for d in divisors(n, generator=True))
    return c.most_common(1)[0][1]
print([a(n) for n in range(1, 88)]) # Michael S. Branicky, Jun 08 2024

a(A326835(n)) = 1.
a(A359563(n)) >= 2.
a(A359565(n)) >= 3.
a(2*n) >= 2.
a(p) = 2 for an odd prime p.
a(m*n) >= a(n) for all m > 1.

A032195 Number of necklaces with 10 black beads and n-10 white beads.

Original entry on oeis.org

1, 1, 6, 22, 73, 201, 504, 1144, 2438, 4862, 9252, 16796, 29414, 49742, 81752, 130752, 204347, 312455, 468754, 690690, 1001603, 1430715, 2016144, 2804880, 3856892, 5245128, 7060984, 9414328, 12440668, 16301164

Offset: 10

Christian G. Bower

nonn

The g.f. is Z(C_10,x)/x^10, the 10-variate cycle index polynomial for the cyclic group C_10, with substitution x[i]->1/(1-x^i), i=1,...,10. By Polya enumeration, a(n+10) is the number of cyclically inequivalent 10-necklaces whose 10 beads are labeled with nonnegative integers such that the sum of labels is n, for n=0,1,2,... See A102190 for Z(C_10,x). See the comment in A032191 on the equivalence of this problem with the one given in the `Name' line. - Wolfdieter Lang, Feb 15 2005

C. G. Bower, Transforms (2)
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Index entries for sequences related to necklaces
Index entries for linear recurrences with constant coefficients, signature (4,-2,-12,17,9,-32,10,29,-29,-9,28,-7,-5,-5,-7,28,-9,-29,29,10,-32,9,17,-12,-2,4,-1).

Column k=10 of A047996.

Cf. A004526, A007997, A008610, A008646, A032191, A032192, A032193, A032194.

k = 10; Table[Apply[Plus, Map[EulerPhi[ # ]Binomial[n/#, k/# ] &, Divisors[GCD[n, k]]]]/n, {n, k, 30}] (* Robert A. Russell, Sep 27 2004 *)

"CIK[ 10 ]" (necklace, indistinct, unlabeled, 10 parts) transform of 1, 1, 1, 1...
G.f.: (x^10)*(1-3*x+4*x^2+12*x^3-8*x^4-x^5+31*x^6-4*x^8+16*x^9 +11*x^10 +3*x^11+8*x^12+4*x^13+4*x^14+x^15+x^16) /((1-x)^4*(1-x^2)^4 *(1-x^5)*(1-x^10)).
G.f.: (1/10)*x^10*(1/(1 - x)^10 + 1/(1 - x^2)^5 + 4/(1 - x^5)^2 + 4/(1 - x^10)^1). - Herbert Kociemba, Oct 22 2016

cp's OEIS Frontend

A032193 Number of necklaces with 8 black beads and n-8 white beads.

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A328858 Numbers with a record number of distinct values of the Euler totient function applied to their divisors (A319696).

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A032194 Number of necklaces with 9 black beads and n-9 white beads.

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A212357 Coefficients for the cycle index polynomial for the cyclic group C_n multiplied by n, n>=1, read as partition polynomial.

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A328859 Indices k of records of low value in the ratios A319696(k)/A000005(k) between the number of distinct values of the Euler totient function applied to the divisors of k and the number of divisors of k.

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A373527 Odd numbers k such that k and k+2 both have at least two divisors with the same value of the Euler totient function (A000010).

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A373528 Odd numbers k such that k, k+2 and k+4 all have at least two divisors with the same value of the Euler totient function (A000010).

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A373529 Numbers k such that k and k+1 both have at least three divisors with the same value of the Euler totient function (A000010).

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A373531 a(n) is the maximum number of divisors of n with an equal value of the Euler totient function (A000010).