A074942 -id:A074942 - cp's OEIS frontend

A088232 Numbers k such that 3 does not divide phi(k).

1, 2, 3, 4, 5, 6, 8, 10, 11, 12, 15, 16, 17, 20, 22, 23, 24, 25, 29, 30, 32, 33, 34, 40, 41, 44, 46, 47, 48, 50, 51, 53, 55, 58, 59, 60, 64, 66, 68, 69, 71, 75, 80, 82, 83, 85, 87, 88, 89, 92, 94, 96, 100, 101, 102, 106, 107, 110, 113, 115, 116, 118, 120, 121, 123, 125, 128

Offset: 1

Yuval Dekel (dekelyuval(AT)hotmail.com), Nov 03 2003

nonn

n such that the congruence x^3 == 1 mod(n) has only the trivial solution x=1 i.e. A060839(n) = 1 . Complement of sequence A066498.
Let U(n) be the group of positive integers coprime to n under mod n multiplication.  Let U(n)^3 = {x^3: x is an element of U(n)}.  These are the n such that U(n) = U(n)^3. - Geoffrey Critzer, Jun 07 2015
In other words, numbers divisible neither by 9 nor by any primes of the form 6k+1. - Ivan Neretin, Sep 03 2015
The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.
Kevin Ford, Florian Luca, and Pieter Moree, Values of the Euler phi-function not divisible by a given odd prime, and the distribution of Euler-Kronecker constants for cyclotomic fields, arXiv:1108.3805 [math.NT], 2011-2012.

Cf. A000010, A066498 (complement).
Positions of 1's in A060839, of 0's in A354099, of nonzeros in A074942.
Cf. also A329963.

select(t -> numtheory:-phi(t) mod 3 <> 0, [$1..1000]); # Robert Israel, Sep 04 2015

Prepend[Position[Table[Union[Select[Range[n], CoprimeQ[#, n] &]] ==
     Union[Mod[Select[Range[n], CoprimeQ[#, n] &]^3, n]], {n, 1,155}], True], 1] // Flatten (* Geoffrey Critzer, Jun 07 2015 *)
Select[Range[140],!Divisible[EulerPhi[#],3]&] (* Harvey P. Dale, Sep 23 2017 *)

is(n)=eulerphi(n)%3 \\ Charles R Greathouse IV, Feb 04 2013

a(n) ~ k n sqrt(log(n)) for some constant k. k appears to be around 1.08. [Charles R Greathouse IV, Feb 14 2012]

More terms from Ray Chandler, Nov 05 2003

A084300 a(n) = phi(n) mod 6.

Original entry on oeis.org

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

Offset: 1

Labos Elemer, Jun 02 2003

Antti Karttunen, Table of n, a(n) for n = 1..10000

Cf. A000010, A010875, A054763, A074942, A084299, A084301.

a[n_] := Mod[EulerPhi[n], 6]; Array[a, 100] (* Amiram Eldar, Aug 17 2024 *)

A084300(n) = (eulerphi(n)%6); \\ Antti Karttunen, Aug 22 2017

a(n) = A000010(n) mod 6.

a(n) = A010875(A000010(n)). - Amiram Eldar, Aug 17 2024

A353768 a(n) = phi(n) mod 4; Euler totient function reduced modulo 4.

Original entry on oeis.org

1, 1, 2, 2, 0, 2, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 2, 2, 0, 0, 2, 2, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 2, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 2, 0, 2, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 0, 2, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 2, 0, 0, 2

Offset: 1

Antti Karttunen, May 15 2022

nonn

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

Cf. A000010, A010873, A066499 (positions of 2's), A172019 (of 0's).

Cf. also A074942, A261872, A084300, and also A105824.

PARI
```
A353768(n) = (eulerphi(n)%4);
```

a(n) = A010873(A000010(n)).

A354099 The 3-adic valuation of Euler totient function phi.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 0, 1, 0, 0, 0, 1, 1, 0, 0, 0, 1, 2, 0, 1, 0, 0, 0, 0, 1, 2, 1, 0, 0, 1, 0, 0, 0, 1, 1, 2, 2, 1, 0, 0, 1, 1, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 2, 0, 1, 2, 0, 0, 0, 1, 1, 2, 0, 1, 0, 1, 0, 0, 1, 0, 1, 2, 2, 0, 2, 1, 1, 1, 0, 3, 0, 0, 1, 0, 1, 0, 0, 0, 1, 2, 0, 1, 0, 2, 0, 1, 1, 1, 0, 0, 0, 1, 1, 1

Offset: 1

Antti Karttunen, May 17 2022

nonn

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

Cf. A000010, A007949, A074942.
Cf. A088232 (positions of zeros), A066498 (of terms > 0).
Cf. also A354100.

a[n_] := IntegerExponent[EulerPhi[n], 3]; Array[a, 100] (* Amiram Eldar, May 17 2022 *)

PARI
```
A354099(n) = valuation(eulerphi(n),3);
```

A354099(n) = { my(f=factor(n)); sum(k=1,#f~,valuation((f[k,1]-1)*(f[k,1]^(f[k,2]-1)), 3)); }; \\ Demonstrates the additivity.

a(n) = A007949(A000010(n)).

Additive with a(p^e) = A007949((p-1)*p^(e-1)).

A261872 a(n) = phi(n) mod 5, where phi is the Euler totient function.

Original entry on oeis.org

1, 1, 2, 2, 4, 2, 1, 4, 1, 4, 0, 4, 2, 1, 3, 3, 1, 1, 3, 3, 2, 0, 2, 3, 0, 2, 3, 2, 3, 3, 0, 1, 0, 1, 4, 2, 1, 3, 4, 1, 0, 2, 2, 0, 4, 2, 1, 1, 2, 0, 2, 4, 2, 3, 0, 4, 1, 3, 3, 1, 0, 0, 1, 2, 3, 0, 1, 2, 4, 4, 0, 4, 2, 1, 0, 1, 0, 4, 3, 2, 4, 0, 2, 4, 4, 2, 1, 0, 3, 4, 2, 4, 0, 1, 2, 2, 1, 2, 0, 0, 0, 2, 2, 3, 3

Offset: 1

Vincenzo Librandi, Sep 04 2015

Antti Karttunen, Table of n, a(n) for n = 1..16384

Cf. A000010, A010874, A074942, A084300.

Magma
```
[EulerPhi(n) mod 5: n in [1..110]];
```
Mathematica
```
Table[Mod[EulerPhi[n], 5], {n, 110}]
```

a(n) = eulerphi(n) % 5; \\ Michel Marcus, Sep 05 2015

a(n) = A000010(n) mod 5 = A010874(A000010(n)).

More terms from Antti Karttunen, Dec 04 2017

A354094 a(n) = phi(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and phi is Euler totient function.

Original entry on oeis.org

1, 4, 2, 20, 10, 8, 6, 100, 6, 40, 16, 40, 12, 24, 20, 500, 22, 24, 18, 200, 12, 64, 28, 200, 110, 48, 18, 120, 40, 80, 30, 2500, 32, 88, 60, 120, 36, 72, 24, 1000, 46, 48, 42, 320, 60, 112, 52, 1000, 42, 440, 44, 240, 58, 72, 160, 600, 36, 160, 70, 400, 60, 120, 36, 12500, 120, 128, 66, 440, 56, 240, 82, 600, 72

Offset: 1

Antti Karttunen, May 17 2022

Index entries for sequences computed from indices in prime factorization

Möbius transform of A354091.
Cf. A000010, A003627, A007949, A008683, A010872, A074942, A354099.
Cf. also A003972.

A354094(n) = { my(f=factor(n)); for(k=1,#f~, if(2==(f[k,1]%3), for(i=1+primepi(f[k,1]),oo,if(2==(prime(i)%3), f[k,1]=prime(i); break)))); eulerphi(factorback(f)); };

Multiplicative with a(p^e) = (q-1) * q^(e-1) where q = A003627(1+n) if p = A003627(n), otherwise q = p.
a(n) = A000010(A354091(n)).
a(n) = Sum_{d|n} A008683(n/d) * A354091(d).
For all n >= 1, A010872(a(n)) = A010872(A000010(n)) = A074942(n).
For all n >= 1, A007949(a(n)) = A007949(A000010(n)) = A354099(n).

cp's OEIS Frontend

A088232 Numbers k such that 3 does not divide phi(k).

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A084300 a(n) = phi(n) mod 6.

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A353768 a(n) = phi(n) mod 4; Euler totient function reduced modulo 4.

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A354099 The 3-adic valuation of Euler totient function phi.

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A261872 a(n) = phi(n) mod 5, where phi is the Euler totient function.

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Magma

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A354094 a(n) = phi(A354091(n)), where A354091 is fully multiplicative prime shift which replaces the primes of the form 3k+2 by the next larger such prime, while other primes stay as they are, and phi is Euler totient function.

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