A009262 -id:A009262 - 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

A036689 Product of a prime and the previous number.

Original entry on oeis.org

2, 6, 20, 42, 110, 156, 272, 342, 506, 812, 930, 1332, 1640, 1806, 2162, 2756, 3422, 3660, 4422, 4970, 5256, 6162, 6806, 7832, 9312, 10100, 10506, 11342, 11772, 12656, 16002, 17030, 18632, 19182, 22052, 22650, 24492, 26406, 27722, 29756, 31862, 32580, 36290, 37056, 38612, 39402, 44310

Offset: 1

Felice Russo

Records in A002618. - Artur Jasinski, Jan 23 2008

Also records in A174857. - Vladimir Shevelev, Mar 31 2010

			2*1, 3*2, 5*4, 7*6, 11*10, 13*12, 17*16, ...

Reinhard Zumkeller, Table of n, a(n) for n = 1..10000
Index to sequences related to prime powers

Cf. A000040, A001248, A002618, A005596, A053650, A053192, A053193, A053650, A082695, A117495, A136141.
Twice the terms of A008837.
Subsequence of A002378 (oblong numbers).
Column 1 of A257251. (Row 1 of A257252.)
Column 2 of A379010.

a036689 n = a036689_list !! (n-1)
a036689_list = zipWith (*) a000040_list $ map pred a000040_list
-- Reinhard Zumkeller, Sep 17 2011

[ n*(n-1): n in PrimesUpTo(220) ]; // Bruno Berselli, Apr 11 2011

A036689 := proc(n) local p ; p := ithprime(n) ; p*(p-1) ; end proc: # R. J. Mathar, Apr 11 2011

Table[Prime[n] EulerPhi[Prime[n]], {n, 100}] (* Artur Jasinski, Jan 23 2008 *)
Table[Prime[n] (Prime[n] - 1), {n, 1, 50}] (* Bruno Berselli, Apr 22 2014 *)
#(#-1)&/@Prime[Range[50]] (* Harvey P. Dale, Sep 08 2019 *)

forprime(p=2,1e3,print1(p^2-p", ")) \\ Charles R Greathouse IV, Jun 10 2011

A036689(n) = ((p->(p-1)*p)(prime(n))); \\ Antti Karttunen, Dec 14 2024

(define (A036689 n) (* (A000040 n) (- (A000040 n) 1))) ;; Antti Karttunen, May 01 2015

a(n) = prime(n) * (prime(n) - 1).
a(n) = phi(prime(n)^2) = A000010(A001248(n)).
a(n) = prime(n) * phi(prime(n)). - Artur Jasinski, Jan 23 2008
From Reinhard Zumkeller, Sep 17 2011: (Start)
a(n) = A000040(n) * A006093(n) = A001248(n) - A000040(n).
A006530(a(n)) = A000040(n). (End)
a(n) = A009262(prime(n)). - Enrique Pérez Herrero, May 12 2012
a(n) = prime(n)! mod (prime(n)^2). - J. M. Bergot, Apr 10 2014
a(n) = 2*A008837(n). - Antti Karttunen, May 01 2015
Sum_{n>=1} 1/a(n) = A136141. - Amiram Eldar, Nov 09 2020
From Amiram Eldar, Jan 23 2021: (Start)
Product_{n>=1} (1 + 1/a(n)) = zeta(2)*zeta(3)/zeta(6) (A082695).
Product_{n>=1} (1 - 1/a(n)) = A005596. (End)

Deleted two incorrect comments. - N. J. A. Sloane, May 07 2020

A174824 a(n) = period of the sequence {m^m, m >= 1} modulo n.

Original entry on oeis.org

1, 2, 6, 4, 20, 6, 42, 8, 18, 20, 110, 12, 156, 42, 60, 16, 272, 18, 342, 20, 42, 110, 506, 24, 100, 156, 54, 84, 812, 60, 930, 32, 330, 272, 420, 36, 1332, 342, 156, 40, 1640, 42, 1806, 220, 180, 506, 2162, 48, 294, 100, 816, 156, 2756, 54, 220, 168, 342

Offset: 1

Franklin T. Adams-Watters, Mar 30 2010

nonn

This is a divisibility sequence: if n divides m, a(n) divides a(m).

We have the equality n = a(n) for numbers n in A124240, which is related to Carmichael's function (A002322). The largest values of a(n) occur when n is prime, in which case a(n) = n*(n-1). - T. D. Noe, Feb 21 2014

			For n=3, 1^1 == 1 (mod 3), 2^2 == 1 (mod 3), 3^3 == 0 (mod 3), etc. The sequence of residues 1, 1, 0, 1, 2, 0, 1, 1, 0, ... has period 6, so a(3) = 6. - _Michael B. Porter_, Mar 13 2018

T. D. Noe, Table of n, a(n) for n = 1..1000
José María Grau and Antonio M. Oller-Marcén, On the last digit and the last non-zero digit of n^n in base b, arXiv:1203.4066 [math.NT], 2012. (See page 3)
Index to divisibility sequences

Cf. A000312, A009262, A127699.

Cf. A002322, A124240, A268336.

Table[LCM[n, CarmichaelLambda[n]], {n, 100}] (* T. D. Noe, Feb 20 2014 *)

a(n)=local(ps);ps=factor(n)[,1]~;for(k=1,#ps,n=lcm(n,ps[k]-1));n

a(n) = lcm(n, lcm(znstar(n)[2])); \\ Michel Marcus, Mar 18 2016; corrected by Michel Marcus, Nov 13 2019

apply( {A174824(n)=lcm(lcm([p-1|p<-factor(n)[,1]]),n)}, [1..99]) \\ [...] = znstar(n)[2], but 3x faster. - M. F. Hasler, Nov 13 2019

a(n) = lcm(n, A173614(n)) = lcm(n, A002322(n)) = lcm(n, A011773(n)).
If n and m are relatively prime, a(n*m) = lcm(a(n), a(m)); a(p^k) = (p-1)*p^k for p prime and k > 0.
a(n) = n*A268336(n). - M. F. Hasler, Nov 13 2019

A052106 a(n) = lcm(n, n - phi(n)).

Original entry on oeis.org

0, 2, 3, 4, 5, 12, 7, 8, 9, 30, 11, 24, 13, 56, 105, 16, 17, 36, 19, 60, 63, 132, 23, 48, 25, 182, 27, 112, 29, 330, 31, 32, 429, 306, 385, 72, 37, 380, 195, 120, 41, 210, 43, 264, 315, 552, 47, 96, 49, 150, 969, 364, 53, 108, 165, 224, 399, 870, 59, 660, 61, 992, 189

Offset: 1

Labos Elemer, Jan 20 2000

nonn

See also A009195, A003277, A050384 when totient and cototient give results identical to each other. This sequence is not identical to A009262.

a(n) = n iff n is in A246655. - Ivan Neretin, May 29 2016

			For n=255, phi(n)=128, cototient(255) = 255 - 128 = 127, a(255) = lcm(255,127) = 32385, while A009262(255) = lcm(255,phi(255)) = 128*255 = 32640;
for n=72, phi(72)=24, A051953(72) = 72 - 24 = 48, a(72) = lcm(72,48) = 144, while A009262(72) = lcm(72,24) = 72.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000010, A009262, A051953.

Table[LCM[n, n - EulerPhi[n]], {n, 63}] (* Ivan Neretin, May 29 2016 *)

a(n) = lcm(n, A051953(n)).

A052100 a(n) = lcm(n, phi(n), n - phi(n)).

Original entry on oeis.org

0, 2, 6, 4, 20, 12, 42, 8, 18, 60, 110, 24, 156, 168, 840, 16, 272, 36, 342, 120, 252, 660, 506, 48, 100, 1092, 54, 336, 812, 1320, 930, 32, 8580, 2448, 9240, 72, 1332, 3420, 1560, 240, 1640, 420, 1806, 1320, 2520, 6072, 2162, 96, 294, 300, 31008, 2184, 2756

Offset: 1

Labos Elemer, Jan 20 2000

nonn

If n is a power of a prime p, then a(n) = n*(p-1). - Robert Israel, May 20 2015

			For n=72, phi(72)=24, cototient(72)=48, a(72) = lcm(72,24,48) = 144.
For n=255, phi(255)=128, cototient(255)=127, a(255) = lcm(255,128,127) = 4145280.

Ivan Neretin, Table of n, a(n) for n = 1..10000

Cf. A000010, A051953, A009262.

seq(ilcm(n, numtheory:-phi(n),n - numtheory:-phi(n)), n=1..100); # Robert Israel, May 20 2015

Table[LCM[n, EulerPhi[n], n - EulerPhi[n]], {n, 53}] (* Ivan Neretin, May 20 2015 *)

a(n) = lcm(n, A000010(n), A051953(n)).
For n=p prime, phi(p)=p-1, cototient(p)=p-1, a(p)=p(p-1)=A009262(p).
a(n) = n*A000010(n)*A051953(n)/A009195(n)^2. - Robert Israel, May 20 2015

A058251 LCM of n-th primorial number and its Euler totient.

Original entry on oeis.org

1, 2, 6, 120, 1680, 36960, 5765760, 1568286720, 536354058240, 24672286679040, 2861985254768640, 2661646286934835200, 3545312854197200486400, 5814313080883408797696000, 10500649424075436288638976000

Offset: 0

Labos Elemer, Dec 05 2000

nonn

			a(6) = LCM(30030,5760) = 5765760.

Cf. A002110, A000010, A005867, A009262.

[seq(ilcm(product(ithprime(k), k=1..m), product(ithprime(k)-1, k=1..m)), m=1..20)];

LCM[#,EulerPhi[#]]&/@Rest[FoldList[Times,1,Prime[Range[15]]]] (* Harvey P. Dale, Nov 29 2011 *)

P(n) = prod(k=1, n, prime(k)); \\ A002110
a(n) = my(p= P(n)); lcm(p, eulerphi(p)); \\ Michel Marcus, Apr 27 2022

a(n) = LCM(A002110(n), A000010(A002110(n))) = LCM(A002110(n), A005867(n)).

a(n) = A009262(A002110(n)). - Michel Marcus, Apr 27 2022

a(0)=1 inserted by Jamie Morken, Apr 27 2022

A328651 Composite k for which lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)).

Original entry on oeis.org

135, 546, 672, 9585, 24570, 51510, 63855, 190008, 251370, 323730, 372438, 486180, 510570, 723550, 819000, 1058910, 1282365, 1284192, 1356030, 3506390, 5416200, 5604480, 6298625, 15593760, 17813250, 18009000, 20740590, 26759370, 27027000, 27081000, 29795040

Offset: 1

Marius A. Burtea, Oct 23 2019

nonn

Composite numbers k verifying equation A009230(k) + A009262(k) = A009242(k).
For any prime number p >= 3 the equality lcm(k, phi(k)) + lcm(k, tau(k)) = lcm(k, sigma(k)) is satisfied.
The sequence terms are the composite numbers for which the equality is true.

			For k = 135 = 3^3 * 5, tau(k) = 4 * 2 = 2^3, phi(k) = 2 * 3^2 * 4 = 2^3 * 3^2 , sigma(k) = 2^4 * 3 * 5, lcm(k, tau(k)) + lcm(k, phi(k)) =  2^3 * 3^3 * 5 + 2^3 * 3^3 * 5 = 2^4 * 3^3 * 5 and lcm(k, sigma(k)) = lcm(3^3 * 5, 2^4 * 3 * 5) = 2^4 * 3^3 * 5.

Cf. A000005, A000010, A000203, A009242, A009230, A009262, A326416.

[k: k in [1..6000000]| not IsPrime(k) and Lcm(k,NumberOfDivisors(k))+Lcm(k,EulerPhi(k)) eq Lcm(k,SumOfDivisors(k))];

aQ[n_] := CompositeQ[n] && LCM[n, EulerPhi[n]] + LCM[n, DivisorSigma[0, n]] == LCM[n, DivisorSigma[1, n]]; Select[Range[3*10^6], aQ] (* Amiram Eldar, Oct 23 2019 *)

isok(k) = !isprime(k) && (lcm(k, numdiv(k)) + lcm(k, eulerphi(k)) == lcm(k, sigma(k))); \\ Michel Marcus, Oct 24 2019

cp's OEIS Frontend

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

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A036689 Product of a prime and the previous number.

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A174824 a(n) = period of the sequence {m^m, m >= 1} modulo n.

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A052106 a(n) = lcm(n, n - phi(n)).

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A052100 a(n) = lcm(n, phi(n), n - phi(n)).

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A058251 LCM of n-th primorial number and its Euler totient.

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