A002618 -id:A002618 - cp's OEIS frontend

A202535 a(n) = nphi(n)abs( mobius(n) ).

1, 2, 6, 0, 20, 12, 42, 0, 0, 40, 110, 0, 156, 84, 120, 0, 272, 0, 342, 0, 252, 220, 506, 0, 0, 312, 0, 0, 812, 240, 930, 0, 660, 544, 840, 0, 1332, 684, 936, 0, 1640, 504, 1806, 0, 0, 1012, 2162, 0, 0, 0, 1632, 0, 2756, 0, 2200, 0, 2052, 1624, 3422

Offset: 1

R. J. Mathar, Dec 20 2011

The inverse Mobius transform is b(n>=1) = 1, 3, 7, 3, 21, 21, 43, 3,7, 63, 11, 21,...., multiplicative with b(p^e) = A002061(p), e>=1 (see A119959). - R. J. Mathar

a(n) > 0 only when n is squarefree. - Alonso del Arte, Dec 20 2011

			a(5) = 20 because 5 * phi(5) * |mu(5)| = 5 * 4 * |(-1)| = 20.

T. D. Noe, Table of n, a(n) for n = 1..1000

Cf. A079579.

Table[n EulerPhi[n] Abs[MoebiusMu[n]], {n, 60}] (* Alonso del Arte, Dec 20 2011 *)
f[p_, e_] := If[e == 1, (p-1)*p, 0]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 26 2020 *)

a(n)=n*eulerphi(n)*abs(moebius(n)) \\ Charles R Greathouse IV, Dec 20 2011

for(n=1, 100, print1(direuler(p=2, n, (1 - p*X + p^2*X))[n], ", ")) \\ Vaclav Kotesovec, Jun 24 2020

a(n) = A002618(n) *A008966(n).
Multiplicative with a(p^e) = (p-1)*p if e=1, a(p^e)=0 if e>1.
Dirichlet g.f.: Sum_(n>=1) a(n)/n^s = Product_{primes p} (1-p^(1-s)+p^(2-s)).
From Vaclav Kotesovec, Jun 24 2020: (Start)
Dirichlet g.f.: zeta(s-2)*Product_{primes p} (1 + p^(3-2*s) - p^(4-2*s) - p^(1-s)).
Sum_{k=1..n} a(k) ~ c * n^3, where c = A065464/3 = 0.142749835... (End)

A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

Original entry on oeis.org

0, 1, 4, 6, 18, 11, 45, 36, 66, 50, 150, 68, 234, 132, 192, 216, 488, 198, 648, 312, 510, 460, 1089, 420, 1140, 732, 1161, 822, 2044, 616, 2430, 1376, 1810, 1528, 2400, 1260, 3942, 2052, 2880, 2008, 5260, 1644, 5943, 2950, 3672, 3509, 7567, 2736, 7497, 3670

Offset: 1

Zhi-Wei Sun, Oct 14 2015

nonn

Conjecture: (i) All the terms of this sequence are pairwise distinct.
(ii) All the numbers phi(n)*pi(n*(n-1)) (n = 1,2,3,...) are pairwise distinct.
(iii) All the numbers phi(n^2)*pi(n^2) = n*phi(n)*pi(n^2) (n = 1,2,3,...) are pairwise distinct.
We have checked this conjecture via Mathematica. For example, we have verified that a(n) (n = 1..4*10^5) are indeed pairwise distinct.
See also A263325 for a similar conjecture.

			a(1) = 0 since pi(1^2)*phi(1)/2 = 0*1/2 = 0.
a(2) = 1 since pi(2^2)*phi(2)/2 = 2*1/2 = 1.
a(3) = 4 since pi(3^2)*phi(3)/2 = 4*2/2 = 4.

Zhi-Wei Sun, Table of n, a(n) for n = 1..10000

Cf. A000010, A000290, A000720, A002618, A038107, A263317, A263321, A263325.

[#PrimesUpTo(n^2)*EulerPhi(n)/2: n in [1..80]]; // Vincenzo Librandi, Oct 15 2015

a[n_]:=a[n]=PrimePi[n^2]*EulerPhi[n]/2
Do[Print[n," ",a[n]],{n,1,50}]

a(n) = primepi(n^2)*eulerphi(n)/2; \\ Michel Marcus, Oct 15 2015

A335319 Decimal expansion of Sum_{n>=2} (-1)^n/(n*phi(n)), where phi(n) is the Euler totient function A000010.

Original entry on oeis.org

5, 5, 9, 2, 2, 8, 6, 8, 0, 7, 1, 2, 4, 2, 8, 0, 4, 2, 4, 2, 5, 4, 3, 4, 3, 3, 6, 7, 0, 3, 9, 8, 2, 0, 6, 7, 4, 8, 6, 5, 6, 5, 3, 6, 1, 2, 4, 2, 4, 2, 8, 2, 7, 3, 1, 6, 5, 9, 0, 0, 8, 9, 1, 0, 2, 5, 6, 6, 6, 2, 2, 6, 3, 7, 6, 2, 9, 4, 6, 0, 9, 0, 0, 4, 8, 5, 4

Offset: 0

Hugo Pfoertner, May 31 2020

The formula section of A000010 provides the following conjecture: Sum_{i>=2} (-1)^i/(i*phi(i)) exists and is approximately 0.558. - Orges Leka (oleka(AT)students.uni-mainz.de), Dec 23 2004

A more accurate value of the conjectured limit is provided.

			0.5592286807124280424254343367039820674865653612424...

Cf. A000010, A002618, A065484.

1 - prodeulerrat(1 + p/((p-1)^2*(p+1)))/5 \\ Amiram Eldar, Nov 11 2020

Equals 1 - (1/5) * A065484. - Amiram Eldar, Nov 11 2020

More terms from Amiram Eldar, Nov 11 2020

A369779 a(n) = n * Sum_{p|n, p prime} phi(n/p) / p.

Original entry on oeis.org

0, 1, 1, 2, 1, 8, 1, 8, 6, 22, 1, 20, 1, 44, 26, 32, 1, 66, 1, 48, 48, 112, 1, 80, 20, 158, 54, 92, 1, 172, 1, 128, 116, 274, 62, 156, 1, 344, 162, 192, 1, 348, 1, 228, 174, 508, 1, 320, 42, 540, 278, 320, 1, 594, 130, 368, 348, 814, 1, 448, 1, 932, 306, 512, 176

Offset: 1

Wesley Ivan Hurt, Jan 31 2024

Dirichlet convolution of A010051(n) and A002618(n). - Wesley Ivan Hurt, Jul 10 2025

Antti Karttunen, Table of n, a(n) for n = 1..20000

Cf. A000010, A002618, A010051, A057660, A117494, A347104, A369687.

Cf. also A369894, A369907, A369909.

Table[n*DivisorSum[n, EulerPhi[n/#]/# &, PrimeQ[#] &], {n, 100}]

A369779(n) = if(1==n, 0, my(f=factor(n)); n*sum(i=1, #f~, (eulerphi(n/f[i, 1])/f[i,1]))); \\ Antti Karttunen, Jan 23 2025

From Wesley Ivan Hurt, Jul 10 2025: (Start)
a(n) = Sum_{d|n} A010051(d) * A002618(n/d).
a(p^k) = ceiling(p^(2k-2)-p^(2k-3)) for p prime and k>=1. (End)

A046078 Primes of the form n*phi(n)-1 where phi is the Euler function (in order of appearance).

Original entry on oeis.org

5, 7, 19, 11, 41, 31, 53, 109, 47, 83, 127, 271, 107, 251, 191, 499, 311, 811, 239, 929, 659, 839, 431, 683, 503, 2161, 971, 3659, 2267, 3119, 1319, 4421, 4969, 2663, 2999, 1871, 4373, 4871, 6551, 9311, 5939, 10099, 5039, 8423, 11423, 13309, 9839, 16001

Offset: 1

Felice Russo

These are listed in order of increasing n.

T. D. Noe, Table of n, a(n) for n=1..1000

Cf. A059109.

Cf. A002618, A046062

Select[Table[n EulerPhi[n]-1,{n,250}],PrimeQ[#]&] (* Harvey P. Dale, Aug 15 2011 *)

Corrected and extended by Jud McCranie, Jan 03 2001

Name clarified by Michel Marcus, Jul 31 2018

A062955 a(n) = phi(n^2) - phi(n) = (n-1) * phi(n).

Original entry on oeis.org

0, 1, 4, 6, 16, 10, 36, 28, 48, 36, 100, 44, 144, 78, 112, 120, 256, 102, 324, 152, 240, 210, 484, 184, 480, 300, 468, 324, 784, 232, 900, 496, 640, 528, 816, 420, 1296, 666, 912, 624, 1600, 492, 1764, 860, 1056, 990, 2116, 752, 2016, 980, 1600, 1224, 2704

Offset: 1

Jason Earls, Jul 22 2001

nonn

Harry J. Smith, Table of n, a(n) for n = 1..1000

Cf. A000010, A002618.

Table[EulerPhi[n^2]-EulerPhi[n],{n,60}] (* Harvey P. Dale, Aug 22 2012 *)

a(n)=if(n<1,0,eulerphi(n^2)-eulerphi(n))

{ for (n=1, 1000, write("b062955.txt", n, " ", eulerphi(n^2) - eulerphi(n)) ) } \\ Harry J. Smith, Aug 14 2009

a(n) = A002618(n) - A000010(n). - Michel Marcus, Jun 28 2018

A074465 a(n) = gcd(n^2, sigma(n^2), phi(n^2)).

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 39, 1, 1, 21, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 7, 1, 1, 1, 1, 1, 1, 7, 39, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 1, 1, 1, 1, 3, 1, 1, 1, 1, 7, 11, 1, 1, 1, 1

Offset: 1

Labos Elemer, Aug 23 2002

nonn

a(n) is odd because sigma(n^2) is odd;.

			For n=14: gcd(196,399,84) = 7 = a(14).

Antti Karttunen, Table of n, a(n) for n = 1..65537

Cf. A000203, A002618, A065764, A074389, A074466.

Table[Apply[GCD, {w^2, DivisorSigma[1, w^2], EulerPhi[w^2]}], {w, 1, 128}]

A074465(n) = gcd([n^2, sigma(n^2), eulerphi(n^2)]); \\ Antti Karttunen, Sep 07 2018

a(n) = A074389(n^2).

A082953 a(n) = A000252(n) / A070732(n).

Original entry on oeis.org

1, 2, 4, 8, 16, 8, 36, 32, 36, 32, 100, 32, 144, 72, 64, 128, 256, 72, 324, 128, 144, 200, 484, 128, 400, 288, 324, 288, 784, 128, 900, 512, 400, 512, 576, 288, 1296, 648, 576, 512, 1600, 288, 1764, 800, 576, 968, 2116

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), May 26 2003

From Jianing Song, Apr 20 2019: (Start)
a(n) is the number of split complex numbers z = x + yj in a reduced system modulo n where x, y are integers, j^2 = 1; number of solutions to gcd(x^2 - y^2, n)=1 with x, y in [0, n-1].
a(n) is the number of invertible elements in the ring Z_n[x]/(x^2 - 1) with discriminant d = 4, where Z_n is the ring of integers modulo n. (End)

Michael De Vlieger, Table of n, a(n) for n = 1..10000

Cf. A000252, A065464, A070732.
Cf. A000010, A062570, A040001.
Similar sequences: A127473 (size of (Z_n[x]/(x^2 - x))*, d = 1), A002618 ((Z_n[x]/(x^2))*, d = 0), A079458 ((Z_n[x]/(x^2 + 1))*, d = -4), A319445 ((Z_n[x]/(x^2 - x + 1))* or (Z_n[x]/(x^2 + x + 1))*, d = -3).

A082953 := proc(n) numtheory[phi](n)*numtheory[phi](2*n) ; end proc:
seq(A082953(n),n=1..100) ; # R. J. Mathar, Jan 07 2011

Array[Times @@ Map[EulerPhi, {#, 2 #}] &, 47] (* Michael De Vlieger, Apr 21 2019 *)

a(n) = eulerphi(n)*eulerphi(2*n); \\ Michel Marcus, Jun 04 2025

a(n) = phi(n)*phi(2*n) = A000010(n)*A062570(n). - Vladeta Jovovic, May 02 2005
Multiplicative with a(2^e) = 2^(2e-1) and a(p^e) = (p-1)^2*p^(2e-2) for p > 2. - R. J. Mathar, Apr 14 2011
a(n) = phi(n)^2 if n odd; 2*phi(n)^2 if n even, where phi(n) = A000010(n). - Jianing Song, Apr 20 2019
Sum_{k=1..n} a(k) ~ c * n^3, where c = (2/5) * Product_{p prime} (1 - (2*p-1)/p^3) = (2/5) * A065464 =  0.171299... . - Amiram Eldar, Oct 30 2022
a(n) = gcd(n,2)*phi(n)^2 = A040001(n)*A127473(n). - Ridouane Oudra, Jun 04 2025

A124827 Order of Galois groups of irreducible Chebyshev polynomials of order n.

Original entry on oeis.org

1, 2, 6, 8, 20, 12, 42, 16, 54, 40, 110, 48, 156, 84, 120, 64, 272, 108, 342, 160, 252

Offset: 1

Artur Jasinski, Nov 09 2006

All groups belonging to solvable Galois groups.
Very similar sequence is A002618 (disagreement occurred only for Chebyshev polynomials orders 8 and 16).
When the order of an irreducible Chebyshev polynomial is a prime number p, the Galois group is the Frobenius group of order p*(p-1) A036689.
In Magma classification the Galois groups are the following: T1_1, T2_1, T3_2, T4_3, T5_3, T6_3, T7_4, T8_8, T9_10, T11_4, T12_28, T13_6, T14_7, T15_11, T16_144, T17_5, T18_45, T19_6, T20_42, T21_15.
Is a(n) the order of Galois group of the polynomial x^n - 2? If so, then a(n) = n*phi(n) for n not divisible by 8, and n*phi(n)/2 otherwise (see the Math Overflow link below). Under this assumption, a(n) is multiplicative with a(p^e) = p^(2*e-1)*(p-1) for p being an odd prime; a(2) = 2, a(4) = 8, and a(2^e) = 2^(2*e-2) for e >= 3. - Jianing Song, Nov 22 2022

			a(5)=20 because the order of the Galois group of polynomial 16x^5-20x^3+5x-c is 20 (where c is an integer chosen so that the polynomial is irreducible). This transitive group is the Frobenius group F5 of order 20 (also called the metacyclic group M_5) T5_3(20) in Magma classification.

Math Overflow, Galois Group of x^n - 2

Cf. A001710, A000142, A036689, A002618, A036689.

Cf. A127835.

Zx:=PolynomialRing(Integers()); f:=16*x^5-20*x^3+5*x-7; G:=GaloisGroup(f:Old); "Order of group",#G; // Juergen Klueners klueners(AT)math.uni-duesseldorf.de

A137316 Array read by rows: T(n,k) is the number of automorphisms of the k-th group of order n, where the ordering is such that the rows are nondecreasing.

Original entry on oeis.org

1, 1, 2, 2, 6, 4, 2, 6, 6, 4, 8, 8, 24, 168, 6, 48, 4, 20, 10, 4, 12, 12, 12, 24, 12, 6, 42, 8, 8, 16, 16, 16, 32, 32, 32, 32, 48, 64, 96, 192, 192, 20160, 16, 6, 12, 48, 54, 432, 18, 8, 20, 24, 40, 40, 12, 42, 10, 110, 22, 8, 16, 16, 24, 24, 24, 24, 24, 24, 48, 48, 48, 48, 144, 336

Offset: 1

Benoit Jubin, Apr 06 2008, Apr 15 2008

The length of the n-th row is A000001(n).
The largest value of the n-th row is A059773(n).
The number phi(n) = A000010(n) appears in the n-th row.

			The table begins as follows:
   1
   1
   2
   2   6
   4
   2   6
   6
   4   8   8  24 168
   6  48
   4  20
  10
   4  12  12  12  24
  12
   6  42
The first row with two numbers corresponds to the two groups of order 4, the cyclic group Z_4 and the Klein group Z_2 x Z_2, whose automorphism groups are respectively the group (Z_4)^* = Z_2 and the symmetric group S_3.

D. MacHale and R. Sheehy, Finite groups with few automorphisms, Math. Proc. Roy. Irish Acad., 104A(2) (2004), 231--238.

Cf. A064767, A060249, A060817, A062771, A060249, A002618, A061350.

# GAP 4
Print("\n") ;
for o in [ 1 .. 33 ] do
    n := NumberSmallGroups(o) ;
    og := [] ;
    for i in [1 .. n] do
        g := SmallGroup(o,i) ;
        H := AutomorphismGroup(g) ;
        ho := Order(H) ;
        Add(og,ho) ;
    od;
    Sort(og) ;
    Print(og) ;
    Print("\n") ;
od; # R. J. Mathar, Jul 13 2013

cp's OEIS Frontend

A202535 a(n) = n*phi(n)*abs( mobius(n) ).

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A263319 a(n) = pi(n^2)*phi(n)/2, where pi(x) denotes the number of primes not exceeding x, and phi(.) is Euler's totient function given by A000010.

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A335319 Decimal expansion of Sum_{n>=2} (-1)^n/(n*phi(n)), where phi(n) is the Euler totient function A000010.

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A369779 a(n) = n * Sum_{p|n, p prime} phi(n/p) / p.

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A046078 Primes of the form n*phi(n)-1 where phi is the Euler function (in order of appearance).

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A062955 a(n) = phi(n^2) - phi(n) = (n-1) * phi(n).

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A074465 a(n) = gcd(n^2, sigma(n^2), phi(n^2)).

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A202535 a(n) = nphi(n)abs( mobius(n) ).