A047918 -id:A047918 - 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

A002619 Number of 2-colored patterns on an n X n board.

Original entry on oeis.org

1, 1, 2, 3, 8, 24, 108, 640, 4492, 36336, 329900, 3326788, 36846288, 444790512, 5811886656, 81729688428, 1230752346368, 19760413251956, 336967037143596, 6082255029733168, 115852476579940152, 2322315553428424200, 48869596859895986108

Offset: 1

N. J. A. Sloane, Colin Mallows

Also number of orbits in the set of circular permutations (up to rotation) under cyclic permutation of the elements. - Michael Steyer, Oct 06 2001

Moser shows that (1/n^2)*Sum_{d|n} k^d*phi(n/d)^2*(n/d)^d*d! is an integer. Here we have k=1. - Michel Marcus, Nov 02 2012

			n=6: {(123456)}, {(135462), (246513), (351624)} and {(124635), (235146), (346251), (451362), (562413), (613524)} are 3 of the 24 orbits, consisting of 1, 3 and 6 permutations, respectively.

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).
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
K. Yordzhev, On the cardinality of a factor set in the symmetric group. Asian-European Journal of Mathematics, Vol. 7, No. 2 (2014) 1450027, doi: 10.1142/S1793557114500272, ISSN:1793-5571, E-ISSN:1793-7183, Zbl 1298.05035.

T. D. Noe, Table of n, a(n) for n = 1..100
M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, Generalized chord diagrams and weight systems, arXiv:2505.24491 [math.CO], 2025. See Theorem 5.3 p. 24.
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. O. J. Moser, A (modest) generalization of the theorems of Wilson and Fermat, Canad. Math. Bull. 33(1990), pp. 253-256.
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 9.
N. J. A. Sloane, Notes on A002618, A002619, etc.
András Szilárd, A combinatorial generalization of Wilson's theorem, Australasian Journal of Combinatorics, Volume 49 (2011), Pages 265-272. See Theorem 3.c p. 269.
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]
A. Vella, Pattern avoidance in permutations: linear and cyclic orders, Electron. J. Combin. 9 (2002/03), no. 2, #R18, 43 pp.
K. Yordzhev, On the cardinality of a factor set in the symmetric group, arXiv:1410.8408 [math.CO], 2014. See Table 2 p. 14.
Saeed Zakeri, Cyclic Permutations: Degrees and Combinatorial Types, arXiv:1909.03300 [math.DS], 2019. See Table 2 p. 10.

Cf. A002618, A047916, A064852, A064649.
Cf. A000010.
Cf. A000939, A000940, A089066, A262480, A275527 (other classes of permutations under various symmetries).

with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(phi(div[j])^2*(n/div[j])!*div[j]^(n/div[j]),j=1..tau(n))/n^2 end: seq(a(n),n=1..23); # Emeric Deutsch, Aug 23 2005

a[n_] := EulerPhi[#]^2*(n/#)!*#^(n/#)/n^2 & /@ Divisors[n] // Total; a /@ Range[23] (* Jean-François Alcover, Jul 11 2011, after Emeric Deutsch *)

a(n)={sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d)/n^2} \\ Andrew Howroyd, Sep 09 2018

from sympy import totient, factorial, divisors
def A002619(n): return sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True))//n**2 # Chai Wah Wu, Nov 07 2022

a(n) = Sum_{k|n} u(n, k)/(nk), where u(n, k) = A047918(n, k).
a(n) = (1/n^2)*Sum_{d|n} phi(d)^2*(n/d)!*d^(n/d), where phi is Euler's totient function (A000010). - Emeric Deutsch, Aug 23 2005
From Richard L. Ollerton, May 09 2021: (Start)
Let A(n,k) = (1/n^2)*Sum_{d|n} k^d*phi(n/d)^2*(n/d)^d*d!, then:
A(n,k) = (1/n^2)*Sum_{i=1..n} k^gcd(n,i)*phi(n/gcd(n,i))*(n/gcd(n,i))^gcd(n,i)*gcd(n,i)!.
A(n,k) = (1/n^2)*Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))^2*(gcd(n,i))^(n/gcd(n,i))*(n/gcd(n,i))!.
a(n) = A(n,1). (End)

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
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]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

A064649 Row sums of the table A047916.

Original entry on oeis.org

1, 4, 12, 40, 140, 816, 5082, 40800, 363258, 3632880, 39916910, 479052528, 6227020956, 87178936992, 1307674429440, 20922800222848, 355687428096272, 6402373892575992, 121645100408832342, 2432902011892837920

Offset: 1

Antti Karttunen, Oct 04 2001

nonn

Harry J. Smith, Table of n, a(n) for n=1..100

Also n*A061417[n]. Cf. A047918, A002619.

a064649 = sum . a047916_row  -- Reinhard Zumkeller, Mar 19 2014

A064649 := proc(n) local d, s; s := 0; for d in divisors(n) do s := s + phi(n/d)*(n/d)^d*d!; od; RETURN(s); end;

a[n_] := DivisorSum[n, EulerPhi[n/#]*(n/#)^#*#!&]; Array[a, 20] (* Jean-François Alcover, Mar 06 2016 *)

{ for (n=1, 100, a=0; v=divisors(n); for (i=1, length(v), d=v[i]; a+=eulerphi(n/d)*(n/d)^d*d!); write("b064649.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 21 2009

a(n) = sumdiv(n, d, eulerphi(n/d)*(n/d)^d*d!); \\ Michel Marcus, Mar 06 2016

a(n) = Sum_{d|n} phi(n/d)*(n/d)^d*d!. - Michel Marcus, Mar 06 2016

A064852 Number of orbits in A002619 consisting of n permutations.

Original entry on oeis.org

1, 0, 0, 1, 4, 18, 102, 624, 4476, 36248, 329890, 3326054, 36846276, 444783906, 5811885808, 81729607680, 1230752346352, 19760412095328, 336967037143578, 6082255011151724, 115852476579789984, 2322315553090615850, 48869596859895986086, 1077167364116800207968

Offset: 1

Michael Steyer (m.steyer(AT)osram.de), Oct 06 2001

Also, the number of aperiodic oriented cycles on n nodes up to rotation of the nodes. See A324513 for illustrations of aperiodic undirected cycles. - Andrew Howroyd, Aug 16 2019

			n=6: The orbit {(124635)(235146)(346251)(451362)(562413)(613524)} consists of 6 single permutations.

Harry J. Smith, Table of n, a(n) for n = 1..100

Cf. A000010, A002619, A008683, A047918, A324513.

a[n_] := Sum[ MoebiusMu[n/k] * EulerPhi[n/k] * (n/k)^k * (k!/n^2), {k, Divisors[n]}]; Table[a[n], {n, 1, 22}] (* Jean-François Alcover, Jun 26 2012, after PARI *)

for(n=1,23,print(sumdiv(n,d,moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2)))

{ for (n=1, 100, a=sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2); write("b064852.txt", n, " ", a) ) } \\ Harry J. Smith, Sep 28 2009

a(n) = Sum_{k|n} mu(n/k)*phi(n/k)*(n/k)^k*k!/n^2 = A047918(n, n)/n^2.

Corrected and extended by Jason Earls and Vladeta Jovovic, Oct 08 2001

A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n).

Original entry on oeis.org

1, 1, 0, 2, 0, 0, 2, 0, 0, 4, 4, 0, 0, 0, 20, 2, 4, 6, 0, 0, 108, 6, 0, 0, 0, 0, 0, 714, 4, 4, 0, 40, 0, 0, 0, 4992, 6, 0, 30, 0, 0, 0, 0, 0, 40284, 4, 16, 0, 0, 380, 0, 0, 0, 0, 362480, 10, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628790, 4, 8, 60, 312, 0, 3768, 0, 0, 0, 0

Offset: 1

N. J. A. Sloane

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
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]

Divide n-th row of array A047918 by n.

Cf. A002024.

a047919 n k = a047919_tabl !! (n-1) !! (k-1)
a047919_row n = a047919_tabl !! (n-1)
a047919_tabl = zipWith (zipWith div) a047918_tabl a002024_tabl
-- Reinhard Zumkeller, Mar 19 2014

U[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; a[n_, k_] := Sum[If[Divisible[n, k], MoebiusMu[d]*U[n, k/d], 0], {d, Divisors[k]}]; row[n_] := Table[a[n, k], {k, 1, n}]/n; Table[row[n], {n, 1, 12}] // Flatten (* Jean-François Alcover, Nov 21 2012, after A047918 *)

Offset corrected by Reinhard Zumkeller, Mar 19 2014

cp's OEIS Frontend

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

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A002619 Number of 2-colored patterns on an n X n board.

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

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A064649 Row sums of the table A047916.

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A064852 Number of orbits in A002619 consisting of n permutations.

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A047919 Triangular array read by rows: a(n,k) = Sum_{d|k} mu(d)*U(n,k/d)/n if k|n else 0, where U(n,k) = A047916(n,k) (1<=k<=n).

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).