A058161 -id:A058161 - cp's OEIS frontend

A002618 a(n) = n*phi(n).

1, 2, 6, 8, 20, 12, 42, 32, 54, 40, 110, 48, 156, 84, 120, 128, 272, 108, 342, 160, 252, 220, 506, 192, 500, 312, 486, 336, 812, 240, 930, 512, 660, 544, 840, 432, 1332, 684, 936, 640, 1640, 504, 1806, 880, 1080, 1012, 2162, 768, 2058, 1000

Offset: 1

N. J. A. Sloane

Also Euler phi function of n^2.
For n >= 3, a(n) is also the size of the automorphism group of the dihedral group of order 2n. This automorphism group is isomorphic to the group of transformations x -> ax + b, where a, b and x are integers modulo n and a is coprime to n. Its order is n*phi(n). - Ola Veshta (olaveshta(AT)my-deja.com), Mar 18 2001
Order of metacyclic group of polynomial of degree n. - Artur Jasinski, Jan 22 2008
It appears that this sequence gives the number of permutations of 1, 2, 3, ..., n that are arithmetic progressions modulo n. - John W. Layman, Aug 27 2008
The conjecture by Layman is correct. Obviously any such permutation must have an increment that is prime to n, and almost as obvious that any such increment will work, with any starting value; hence phi(n) * n total. - Franklin T. Adams-Watters, Jun 09 2009
Consider the numbers from 1 to n^2 written line by line as an n X n square: a(n) = number of numbers that are coprime to all their horizontal and vertical immediate neighbors. - Reinhard Zumkeller, Apr 12 2011
n -> a(n) is injective: a(m) = a(n) implies m = n. - Franz Vrabec, Dec 12 2012 (See Mathematics Stack Exchange link for a proof.)
a(p) = p*(p-1) a pronic number, see A036689 and A002378. - Fred Daniel Kline, Mar 30 2015
Conjecture: All the rational numbers Sum_{i=j..k} 1/a(i) with 0 < min{2,k} <= j <= k have pairwise distinct fractional parts. - Zhi-Wei Sun, Sep 24 2015
From Jianing Song, Aug 25 2023: (Start)
a(n) is the order of the holomorph (see the Wikipedia link) of the cyclic group of order n. Note that Hol(C_n) and Aut(D_{2n}) are isomorphic unless n = 2, where D_{2n} is the dihedral group of order 2*n. See the Wordpress link.
The odd-indexed terms form a subsequence of A341298: the holomorph of an abelian group of odd order is a complete group. See Theorem 3.2, Page 618 of the W. Peremans link. (End)

			a(4) = 8 since phi(4) = 2 and 4 * 2 = 8.
a(5) = 20 since phi(5) = 4 and 5 * 4 = 20.

Peter Giblin, Primes and Programming: An Introduction to Number Theory with Computing. Cambridge: Cambridge University Press (1993) p. 116, Exercise 1.10.
J. L. Lagrange, Oeuvres, Vol. III Paris 1869.
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).

Michael De Vlieger, Table of n, a(n) for n = 1..10000 (first 1000 terms from T. D. Noe)
Daniel Fischer, answer to Injectivity of the function n times the Euler Totient of n, Mathematics Stack Exchange, October 2013.
Mikhail R. Gabdullin and Vitalii V. Iudelevich, Numbers of the form kf(k), arXiv:2201.09287 [math.NT] (2022).
Dmitry Krachun and Zhi-Wei Sun, Each positive rational number has the form phi(m^2)/phi(n^2), arXiv:2001.03736 [math.HO], 2020. See also The American Mathematical Monthly (2020) Vol. 127, Issue 9, 847-849.
F. Luca and A. O. Munagi, The number of permutations which form arithmetic progressions modulo m, Annals of the Alexandru Ioan Cuza University, 2014, DOI: 10.2478/aicu-2014-0053. [Broken link]
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. Peremans, Completeness of Holomorphs, Nederl. Akad. Wetensch. Indag. Math. Proc. Ser. A, 60. (1957) 608-619.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
Wikipedia, Holomorph.
Wordpress, Automorphisms of dihedral groups.

First column of A047916.
Cf. A002619, A011755 (partial sums), A047918, A000010, A053650, A053191, A053192, A036689, A058161, A009262, A082473 (same terms, sorted into ascending order), A256545, A327172 (a left inverse), A327173, A065484.
Subsequence of A323333.

a002618 n = a000010 n * n  -- Reinhard Zumkeller, Dec 21 2012

[n*EulerPhi(n): n in [1..150]]; // Vincenzo Librandi, Apr 04 2011

with(numtheory):a:=n->phi(n^2): seq(a(n), n=1..50); # Zerinvary Lajos, Oct 07 2007

Table[n EulerPhi[n], {n, 100}] (* Artur Jasinski, Jan 22 2008 *)

numlib::phi(n^2)$ n=1..81 // Zerinvary Lajos, May 13 2008

a(n)=n*eulerphi(n) \\ Charles R Greathouse IV, Nov 20 2012

from sympy import totient as phi
def a(n): return n*phi(n)
print([a(n) for n in range(1, 51)]) # Michael S. Branicky, Mar 16 2022

[euler_phi(n^2) for n in range(1,51)] # Zerinvary Lajos, Jun 06 2009

Multiplicative with a(p^e) = (p-1)*p^(2e-1). - David W. Wilson, Aug 01 2001
Dirichlet g.f.: zeta(s-2)/zeta(s-1). - R. J. Mathar, Feb 09 2011
a(n) = A173557(n) * A102631(n). - R. J. Mathar, Mar 30 2011
From Wolfdieter Lang, May 12 2011: (Start)
a(n)/2 = A023896(n), n >= 2.
a(n)/2 = (1/n) * Sum_{k=1..n-1, gcd(k,n)=1} k, n >= 2 (see A023896 and A076512/A109395). (End)
a(n) = lcm(phi(n^2),n). - Enrique Pérez Herrero, May 11 2012
a(n) = phi(n^2). - Wesley Ivan Hurt, Jun 16 2013
a(n) = A009195(n) * A009262(n). - Michel Marcus, Oct 24 2013
G.f.: -x + 2*Sum_{k>=1} mu(k)*k*x^k/(1 - x^k)^3. - Ilya Gutkovskiy, Jan 03 2017
a(n) = A082473(A327173(n)), A327172(a(n)) = n. -- Antti Karttunen, Sep 29 2019
Sum_{n>=1} 1/a(n) = 2.203856... (A065484). - Amiram Eldar, Sep 30 2019
Define f(x) = #{n <= x: a(n) <= x}. Gabdullin & Iudelevich show that f(x) ~ c*sqrt(x) for c = Product_{p prime} (1 + 1/(p*(p - 1 + sqrt(p^2 - p)))) = 1.3651304521525857... - Charles R Greathouse IV, Mar 16 2022
a(n) = Sum_{d divides n} A001157(d)*A046692(n/d); that is, the Dirichlet convolution of sigma_2(n) and the Dirichlet inverse of sigma_1(n). - Peter Bala, Jan 26 2024

Better description from Labos Elemer, Feb 18 2000

A051625 Number of "labeled" cyclic subgroups of symmetric group S_n.

Original entry on oeis.org

1, 2, 5, 17, 67, 362, 2039, 14170, 109694, 976412, 8921002, 101134244, 1104940280, 13914013024, 191754490412, 2824047042632, 41304021782824, 708492417746000, 11629404776897384, 222093818836736752, 4351196253952132832, 88481681599705382144, 1781763397966126421200

Offset: 1

Vladeta Jovovic

nonn

Number of unordered lists of powers of permutation of length n (equivalent to the definition). - Olivier Gérard, Jul 04 2011

Number of subgroups of S_n with different permutations generated by single permutation (see Mathematica procedure). - Artur Jasinski, Oct 27 2011

			The 5 cyclic subgroups of symmetric group S_3 are: {Id}, the 3 subgroups {Id,(a,b)}, {Id,(b,c)}, {Id,(a,c)} and the Alternating group A_3: <Id, (a,b,c), (a,c,b)>.
The 17 cyclic subgroups of symmetric group S_4 are: {Id}, the 6 subgroups of type <(a,b)>, the 3 subgroups of type <(a,b)(c,d)>, the 4 subgroups of type <(a,b,c)> and the 3 subgroups of type <(a,b,c,d)>. - _Bernard Schott_, Feb 25 2019

V. Jovovic, Some combinatorial characteristics of symmetric and alternating groups (in Russian), Belgrade, 1980, unpublished.

Alois P. Heinz, Table of n, a(n) for n = 1..130
Peter J. Cameron, Andrea Lucchini and Colva M. Roney-Dougal, Generating sets of finite groups, arXiv:1609.06077 [math.GR], 2016.
Adam Chojecki, Paweł Morgen, and Bartosz Kołodziejek, Learning permutation symmetries with gips in R, arXiv:2307.00790 [stat.CO], 2023.
Piotr Graczyk, Hideyuki Ishi, Kołodziejek Bartosz, and Hélène Massam, Model selection in the space of Gaussian models invariant by symmetry, arXiv:2004.03503 [math.ST], 2020.
L. Naughton and G. Pfeiffer, Integer Sequences Realized by the Subgroup Pattern of the Symmetric Group, J. Int. Seq. 16 (2013) #13.5.8.
Index entries for sequences related to groups

Row sums of A074881.

Cf. A000010, A000110, A051636, A058161.

parts:= proc(n,k) option remember;
   if k = 1 then return {[n]} fi;
   `union`(seq(map(t -> [op(t),j], procname(n-j*k,k-1)), j=0..floor(n/k)))
end proc:
F:= n -> add(n!/mul(p[k]!*k^p[k],k=1..nops(p)) / numtheory:-phi(ilcm(op(select(t -> p[t]<>0, [$1..n])))), p = parts(n,n)):
seq(F(n),n=1..30); # Robert Israel, Oct 04 2015

cc = {}; Do[aa = {}; kk = Table[n, {n, 1, ord}]; pp = Permutations[kk]; Do[per17 = {}; AppendTo[per17, pp[[p]]]; run = 0; ile = Length[per17]; min = 1; max = ile; While[ile < ord!, run = run + 1; if = False; Do[Do[vec0 = Table[0, {n, 1, ord}]; Do[vec0[[per17[[k]][[n]]]] = per17[[m]][[n]], {n, 1, ord}]; bp = vec0; If[Position[per17, bp] == {}, ile = ile + 1; Print[ile]; if = True; AppendTo[per17, bp]]; vec0 = Table[0, {n, 1, ord}]; Do[vec0[[per17[[m]][[n]]]] = per17[[k]][[n]], {n, 1, ord}]; bl = vec0; If[Position[per17, bl] == {}, ile = ile + 1; if = True; AppendTo[per17, bl]]; If[ile == ord!, Break[]], {k, 1, max}], {m, min, max}]; If[if == False, Break[], min = max + 1; max = ile]]; AppendTo[aa, Sort[per17]], {p, 1, ord!}]; AppendTo[cc, Length[Union[aa]]], {ord, 1, 7}]; cc (* Artur Jasinski, Oct 27 2011 *)
permcount[v_] := Module[{m=1, s=0, k=0, t}, For[i=1, i <= Length[v], i++, t = v[[i]]; k = If[i>1 && t == v[[i-1]], k+1, 1]; m *= t*k; s += t]; s!/m];
a[n_] := Module[{s = 0}, Do[s += permcount[p]/EulerPhi[LCM @@ p], {p, IntegerPartitions[n]}]; s];
Array[a, 23] (* Jean-François Alcover, Feb 25 2019, after Andrew Howroyd *);
content[li_List] := Table[Count[li, i], {i, Tr[li]}]; Table[Tr[(n!/(Times @@ (Range[Tr[#1]]^content[#1]*content[#1]!)*EulerPhi[LCM @@ Flatten[Position[content[#1], ?Positive]]]) & ) /@ IntegerPartitions[n] ], {n, 23}] (* _Wouter Meeussen, Jan 06 2021 *);

\\ permcount is number of permutations of given type.
permcount(v) = {my(m=1,s=0,k=0,t); for(i=1,#v,t=v[i]; k=if(i>1&&t==v[i-1],k+1,1); m*=t*k;s+=t); s!/m}
a(n)={my(s=0); forpart(p=n, s+=permcount(p)/eulerphi(lcm(Vec(p)))); s} \\ Andrew Howroyd, Jul 03 2018

a(n) = Sum_{pi} n!/(k_1!*1^k_1*k_2!*2^k_2*...*k_n!*n^k_n*phi(lcm{i:k_i != 0})), where pi runs through all partitions k_1+2*k_2+...+n*k_n=n and phi is Euler's function.

A058162 Number of labeled Abelian groups with a fixed identity.

Original entry on oeis.org

1, 1, 1, 4, 6, 60, 120, 1920, 7560, 90720, 362880, 13305600, 39916800, 1037836800, 10897286400, 265686220800, 1307674368000, 66691392768000, 355687428096000, 20274183401472000, 202741834014720000

Offset: 1

Christian G. Bower, Nov 15 2000, Mar 12 2008

nonn

The distinction here between labeled and unlabeled Abelian groups is analogous to the distinction between unlabeled rooted trees (A000081) and labeled rooted trees (A000169).
That is, the number of Cayley tables. - Artur Jasinski, Mar 12 2008
Number of Latin squares in dimension n with first row and first column 1,2,3 ..., n which are associative and commutative (Abelian). Each of these squares is isomorphic with the Cayley table of one of the existed Abelian group in dimension n. - Artur Jasinski, Nov 02 2005. Cf. A111341.

			The 2 unlabeled Abelian groups of order 4 are C4 and C2^2. The 4 labeled Abelian groups whose identity is "0" consist of 3 of type C4 (where the nongenerator can be "2", "3", or "4") and 1 of type C2^2.

Max Alekseyev, Table of n, a(n) for n = 1..100
C. J. Hillar, D. Rhea. Automorphisms of finite Abelian groups. American Mathematical Monthly 114:10 (2007), 917-923. Preprint arXiv:math/0605185 [math.GR]
Index entries for sequences related to groups

Cf. A000688, A058160, A058161, A058163.

a(n) = A034382(n) / n. Formula for A034382 is based on the fundamental theorem of finite Abelian groups and the formula given by Hillar and Rhea (2007).

a(16) and a(21) corrected by Max Alekseyev, Sep 12 2019

A034381 Number of labeled cyclic groups with n elements.

Original entry on oeis.org

1, 2, 3, 12, 30, 360, 840, 10080, 60480, 907200, 3991680, 119750400, 518918400, 14529715200, 163459296000, 2615348736000, 22230464256000, 1067062284288000, 6758061133824000, 304112751022080000, 4257578514309120000, 112400072777760768000, 1175091669949317120000

Offset: 1

Christian G. Bower

This sequence is strictly increasing, since a(n) = n!/phi(n) > n!/n = (n-1)! >= a(n-1) for n >= 2. - Jianing Song, Mar 02 2024

Jianing Song, Table of n, a(n) for n = 1..450
Index entries for sequences related to groups

Cf. A000010, A000142, A034382, A034383, A058161.

a(n) = n! / eulerphi(n) \\ Jianing Song, Mar 02 2024

a(n) = n!/phi(n).

a(n) = A000142(n)/A000010(n) = n*A058161(n).

a(21) onwards from Jianing Song, Mar 02 2024

A137149 a(n) = (prime(n)-2)!.

Original entry on oeis.org

1, 1, 6, 120, 362880, 39916800, 1307674368000, 355687428096000, 51090942171709440000, 10888869450418352160768000000, 8841761993739701954543616000000

Offset: 1

Artur Jasinski, Jan 23 2008

nonn

Old definition was "a(n) = prime(n)!/(prime(n)*EulerPhi(prime(n)))".

Degree of Lagrange resolvent of polynomial prime degree. Ratio: degree of symmetric group of prime order n divided by order metacyclic group of prime order n. For degree of Lagrange resolvent of polynomial not prime degree see A137150.

Vincenzo Librandi, Table of n, a(n) for n = 1..88

Cf. A058161, A137150.

[Factorial(NthPrime(n)-2): n in [1..15]]; // Vincenzo Librandi, May 04 2014

Table[(Prime[n]-2)!, {n, 1, 15}] (* Bruno Berselli, May 04 2014 *)

New name from Vincenzo Librandi, May 04 2014

A137150 Degree of Lagrange resolvent of polynomial of composite degree.

Original entry on oeis.org

1, 3, 60, 1260, 6720, 90720, 9979200, 1037836800, 10897286400, 163459296000, 59281238016000, 15205637551104000, 202741834014720000, 5109094217170944000, 3231502092360622080000, 31022420086661971968000

Offset: 1

Artur Jasinski, Jan 23 2008

nonn

Ratio: degree of symmetric group of composite order n divided by order metacyclic group of composite order n.

Vincenzo Librandi, Table of n, a(n) for n = 1..300

Cf. A058161, A137149.

a = {}; Do[If[PrimeQ[n],[null], AppendTo[a, n!/(n EulerPhi[n])]], {n, 1, 30}]; a
With[{nn=30},#!/(# EulerPhi[#])&/@Complement[Range[nn],Prime[Range[ PrimePi[ nn]]]]] (* Harvey P. Dale, Jul 05 2014 *)

a(n) = n!/(n EulerPhi[n]) for composite n A058161 = A137149 + A137150.

A336412 Number of labeled dihedral groups with a fixed identity.

Original entry on oeis.org

1, 1, 20, 630, 18144, 3326400, 148262400, 40864824000, 6586804224000, 3041127510220800, 464463110651904000, 538583682060103680000, 99430833611096064000000, 129629398219266097152000000, 73681349947830849621196800000, 64240926985765022013480960000000

Offset: 1

Dan Eilers, Jul 20 2020

nonn

a(n) is the number of dihedral groups of order 2n with a fixed identity, or equivalently the number of reduced Latin squares of order 2n that can be viewed as the Cayley table of D_{2n}, by adding a border that matches the first row and column. The reduced Latin squares differ from each other by a permutation of their symbols. Two such Latin squares that differ by a permutation of their symbols have been called isoplanar by Bailey (1984), cited by Nilrat and Praeger (1988), cited by Denes and Keedwell (1991). Latin squares based on dihedral groups are of interest in the stable marriage problem, where Benjamin et al. (1995) exhibited such squares having many stable matchings when viewed as ranking matrices. Two isoplanar Latin squares generally produce a different number of stable matchings, so there is motivation to generate all symbol permutations to find the ones with the most stable matchings.

See comments in A002618 regarding automorphisms of dihedral groups by Ola Veshta and Yaghoub Sharifi. - Dan Eilers, Jun 08 2024

			For n=3 the a(3)=20 isoplanar reduced Latin squares based on the dihedral group of order 6, in lexicographical order, are:
1)             2)             3)             4)             5)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 4 3 6 5    2 1 4 3 6 5    2 1 4 3 6 5    2 1 4 3 6 5    2 1 5 6 3 4
3 5 1 6 2 4    3 5 6 2 4 1    3 6 1 5 4 2    3 6 5 2 1 4    3 4 1 2 6 5
4 6 2 5 1 3    4 6 5 1 3 2    4 5 2 6 3 1    4 5 6 1 2 3    4 3 6 5 1 2
5 3 6 1 4 2    5 3 2 6 1 4    5 4 6 2 1 3    5 4 1 6 3 2    5 6 2 1 4 3
6 4 5 2 3 1    6 4 1 5 2 3    6 3 5 1 2 4    6 3 2 5 4 1    6 5 4 3 2 1
6)             7)             8)             9)             10)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 5 6 3 4    2 1 5 6 3 4    2 1 5 6 3 4    2 1 6 5 4 3    2 1 6 5 4 3
3 4 6 5 2 1    3 6 1 5 4 2    3 6 4 1 2 5    3 4 1 2 6 5    3 4 5 6 1 2
4 3 2 1 6 5    4 5 6 1 2 3    4 5 1 3 6 2    4 3 5 6 2 1    4 3 2 1 6 5
5 6 4 3 1 2    5 4 2 3 6 1    5 4 6 2 1 3    5 6 4 3 1 2    5 6 1 2 3 4
6 5 1 2 4 3    6 3 4 2 1 5    6 3 2 5 4 1    6 5 2 1 3 4    6 5 4 3 2 1
11)            12)            13)            14)            15)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 1 6 5 4 3    2 1 6 5 4 3    2 3 1 5 6 4    2 3 1 6 4 5    2 4 5 1 6 3
3 5 1 6 2 4    3 5 4 1 6 2    3 1 2 6 4 5    3 1 2 5 6 4    3 6 1 5 4 2
4 6 5 1 3 2    4 6 1 3 2 5    4 6 5 1 3 2    4 5 6 1 2 3    4 1 6 2 3 5
5 3 4 2 6 1    5 3 2 6 1 4    5 4 6 2 1 3    5 6 4 3 1 2    5 3 2 6 1 4
6 4 2 3 1 5    6 4 5 2 3 1    6 5 4 3 2 1    6 4 5 2 3 1    6 5 4 3 2 1
16)            17)            18)            19)            20)
1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6    1 2 3 4 5 6
2 4 6 1 3 5    2 5 4 6 1 3    2 5 6 3 1 4    2 6 4 5 3 1    2 6 5 3 4 1
3 5 1 6 2 4    3 6 1 5 4 2    3 4 1 2 6 5    3 5 1 6 2 4    3 4 1 2 6 5
4 1 5 2 6 3    4 3 2 1 6 5    4 6 5 1 3 2    4 3 2 1 6 5    4 5 6 1 2 3
5 6 4 3 1 2    5 1 6 3 2 4    5 1 4 6 2 3    5 4 6 2 1 3    5 3 2 6 1 4
6 3 2 5 4 1    6 4 5 2 3 1    6 3 2 5 4 1    6 1 5 3 4 2    6 1 4 5 3 2

Denes, J. and Keedwell, A. D. (1991) Latin Squares New Developments in the Theory and Applications. p. 98.

R. A. Bailey, Quasi-Complete Latin Squares: Construction and Randomization, Journal of the Royal Statistical Society. Series B (Methodological) 46, no. 2 (1984): 330, 323-34.
A. T. Benjamin, C. Converse, and H. A. Krieger, Note. How do I marry thee? Let me count the ways, Discrete Appl. Math. 59 (1995) 285-292.
C. K. Nilrat and C. E. Prager, Complete latin squares: terraces for groups, Ars Combinatoria 24 (1988), 17-29.
Yaghoub Sharifi, Automorphisms of dihedral groups.
E. G. Thurber, Concerning the maximum number of stable matchings in the stable marriage problem, Discrete Mathematics Volume 248, Issue 1-3, 6 April 2002, 195-219.

Cf. A058163 (all groups), A058162 (Abelian groups), A058161 (cyclic groups), A069156 (stable matchings), A002618 (n*phi(n)).

A336412:=List([1..16], n->Factorial(2*n-1)/Size(AutomorphismGroup(DihedralGroup(2*n)))); # Dan Eilers, Jun 08 2024

a(1) = a(2) = 1; a(n>2) = (2*n-1)! / A002618(n). - Dan Eilers, Jun 08 2024

a(8)-a(16) and edited by Dan Eilers, Jun 08 2024

cp's OEIS Frontend

A002618 a(n) = n*phi(n).

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A051625 Number of "labeled" cyclic subgroups of symmetric group S_n.

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A058162 Number of labeled Abelian groups with a fixed identity.

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A034381 Number of labeled cyclic groups with n elements.

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A137149 a(n) = (prime(n)-2)!.

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A137150 Degree of Lagrange resolvent of polynomial of composite degree.

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A336412 Number of labeled dihedral groups with a fixed identity.

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