A066499 -id:A066499 - cp's OEIS frontend

A066498 Numbers k such that 3 divides phi(k).

7, 9, 13, 14, 18, 19, 21, 26, 27, 28, 31, 35, 36, 37, 38, 39, 42, 43, 45, 49, 52, 54, 56, 57, 61, 62, 63, 65, 67, 70, 72, 73, 74, 76, 77, 78, 79, 81, 84, 86, 90, 91, 93, 95, 97, 98, 99, 103, 104, 105, 108, 109, 111, 112, 114, 117, 119, 122, 124, 126, 127, 129, 130, 133

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Numbers k such that x^3 == 1 (mod k) has solutions 1 < x < k.
Terms are multiple of 9 or of a prime of the form 6k+1.
If k is a term of this sequence, then G =  is a non-abelian group of order 3k, where 1 < r < n and r^3 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Mar 21 2021

			If n < 7 then x^3 = 1 (mod n) has no solution 1 < x < n, but {2,4} are solutions to x^3 == 1 (mod 7), hence a(1) = 7.

Harry J. Smith, Table of n, a(n) for n=1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Complement of A088232.
Cf. A000010, A066499, A066500, A066501, A066502.
A007645 gives the primes congruent to 1 mod 3.
Column k=2 of A277915.

Select[Range[150], Divisible[EulerPhi[#], 3]&] (* Harvey P. Dale, Jan 12 2011 *)

isok(k)={ eulerphi(k)%3 == 0 } \\ Harry J. Smith, Feb 18 2010

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Corrected and extended by Ray Chandler, Nov 05 2003

A066502 Numbers k such that 7 divides phi(k).

Original entry on oeis.org

29, 43, 49, 58, 71, 86, 87, 98, 113, 116, 127, 129, 142, 145, 147, 172, 174, 196, 197, 203, 211, 213, 215, 226, 232, 239, 245, 254, 258, 261, 281, 284, 290, 294, 301, 319, 337, 339, 343, 344, 348, 355, 377, 379, 381, 387, 392, 394, 406, 421, 422, 426, 430

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^7 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^7 == 1 (mod k).
If k is a term of this sequence, then G =  is a non-abelian group of order 7k, where 1 < r < n and r^7 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, May 23 2022

			x^7 == 1 (mod k) has solutions 1 < x < k for k = 29, 43, 49, ...

Harry J. Smith, Table of n, a(n) for n = 1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A045465, A066498, A066499, A066500, A066501, A000010.

Column k=4 of A277915.

Select[Range[500],Divisible[EulerPhi[#],7]&] (* Harvey P. Dale, Apr 12 2012 *)

isok(k) = { eulerphi(k)%7 == 0 } \\ Harry J. Smith, Feb 18 2010

a(n) are the numbers generated by 7^2 = 49 and all primes congruent to 1 mod 7 (A045465). Hence sequence gives all k such that k == 0 (mod A045465(n)) for some n > 1 or k == 0 (mod 49).

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

A066500 Numbers k such that 5 divides phi(k).

Original entry on oeis.org

11, 22, 25, 31, 33, 41, 44, 50, 55, 61, 62, 66, 71, 75, 77, 82, 88, 93, 99, 100, 101, 110, 121, 122, 123, 124, 125, 131, 132, 142, 143, 150, 151, 154, 155, 164, 165, 175, 176, 181, 183, 186, 187, 191, 198, 200, 202, 205, 209, 211, 213, 217, 220, 225, 231, 241

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Related to the equation x^5 == 1 (mod k): sequence gives values of k such there are solutions 1 < x < k of x^5 == 1 (mod k).
If k is a term of this sequence, then G =  is a non-abelian group of order 5k, where 1 < r < n and r^5 == 1 (mod k). For example, G can be the subgroup of GL(2, Z_k) generated by x = {{1, 1}, {0, 1}} and y = {{r, 0}, {0, 1}}. - Jianing Song, Sep 17 2019
The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, May 23 2022

			x^5 == 1 (mod 11) has solutions 1 < x < 11, namely {3,4,5,9}.

Harry J. Smith, Table of n, a(n) for n = 1..1000
Robert E. Dressler, A property of the phi and sigma_j functions, Compositio Mathematica, Vol. 31, No. 2 (1975), pp. 115-118.

Cf. A000010, A045453, A066498, A066499, A066501, A066502.

Column k=3 of A277915.

Select[Range[250], Divisible[EulerPhi[#], 5] &] (* Amiram Eldar, May 23 2022 *)

isok(k) = { eulerphi(k)%5 == 0 } \\ Harry J. Smith, Feb 18 2010

a(n) are the numbers generated by 5^2 = 25 and all primes congruent to 1 mod 5 (A045453). Hence sequence gives all k such that k == 0 (mod A045453(n)) for some n > 1 or k == 0 (mod 25).

Simpler definition from Yuval Dekel (dekelyuval(AT)hotmail.com), Oct 25 2003

Extended by Ray Chandler, Nov 06 2003

A066501 Numbers k such that x^6 == 1 (mod(k)) has no solution 1 < x < k-1.

Original entry on oeis.org

1, 2, 3, 4, 5, 6, 10, 11, 17, 22, 23, 25, 29, 34, 41, 46, 47, 50, 53, 58, 59, 71, 82, 83, 89, 94, 101, 106, 107, 113, 118, 121, 125, 131, 137, 142, 149, 166, 167, 173, 178, 179, 191, 197, 202, 214, 226, 227, 233, 239, 242, 250, 251, 257, 262, 263, 269, 274, 281, 289, 293

Offset: 1

Benoit Cloitre, Jan 04 2002

nonn

Cf. A045309, A066498, A066499, A066500, A066502, A000010.

isok(n) = {for (x=2, n-2, if ((Mod(x, n)^6) == Mod(1, n), return (0));); return (1);} \\ Michel Marcus, Nov 20 2013

Sequence consists of the numbers 4, 6 and for all k > 1, A045309(k), 2*A045309(k), A045309(k)^2, 2*A045309(k)^2.

Extended by Ray Chandler, Nov 06 2003

Terms 1, 2 and 3 prepended by Michel Marcus, Nov 20 2013

A172019 Numbers k such that 4 divides phi(k) (i.e., A000010(k)).

Original entry on oeis.org

5, 8, 10, 12, 13, 15, 16, 17, 20, 21, 24, 25, 26, 28, 29, 30, 32, 33, 34, 35, 36, 37, 39, 40, 41, 42, 44, 45, 48, 50, 51, 52, 53, 55, 56, 57, 58, 60, 61, 63, 64, 65, 66, 68, 69, 70, 72, 73, 74, 75, 76, 77, 78, 80, 82, 84, 85, 87, 88, 89, 90, 91, 92, 93, 95, 96, 97, 99, 100, 101

Offset: 1

Giovanni Teofilatto, Jan 22 2010

Complement of A097987.

The asymptotic density of this sequence is 1 (Dressler, 1975). - Amiram Eldar, Feb 12 2021

Rémy Sigrist, 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.

Cf. A000010, A097987, A066499, A066498, A066500, A066502.

Select[Range[200], Mod[EulerPhi[#], 4] == 0 &] (* Geoffrey Critzer, Nov 30 2014 *)

is(n)=my(o=valuation(n, 2), p); (o>1 || !isprimepower(n>>o, &p) || p%4<2) && n>4 \\ Charles R Greathouse IV, Mar 05 2013

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)).

A097987 Numbers k such that 4 does not divide phi(k), where phi is Euler's totient function (A000010).

Original entry on oeis.org

1, 2, 3, 4, 6, 7, 9, 11, 14, 18, 19, 22, 23, 27, 31, 38, 43, 46, 47, 49, 54, 59, 62, 67, 71, 79, 81, 83, 86, 94, 98, 103, 107, 118, 121, 127, 131, 134, 139, 142, 151, 158, 162, 163, 166, 167, 179, 191, 199, 206, 211, 214, 223, 227, 239, 242, 243, 251, 254, 262, 263, 271

Offset: 1

Lekraj Beedassy, Sep 07 2004

nonn

The asymptotic density of this sequence is 0 (Dressler, 1975). - Amiram Eldar, Jul 23 2020

Ivan Neretin, 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.

Essentially the same as A066499.
Cf. A000010.
Complement of A172019.

Select[Range@275, ! Divisible[EulerPhi[#], 4] &] (* Ivan Neretin, Aug 24 2016 *)

is(n)=my(o=valuation(n,2),p); (o<2 && isprimepower(n>>o,&p) && p%4>1) || n<5 \\ Charles R Greathouse IV, Feb 21 2013

a(n)=1, 2, 4, p^k, 2*p^k, with prime p == 3 (mod 4).

Corrected and extended by Vladeta Jovovic, Sep 08 2004

cp's OEIS Frontend

A066498 Numbers k such that 3 divides phi(k).

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A066502 Numbers k such that 7 divides phi(k).

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A066500 Numbers k such that 5 divides phi(k).

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A066501 Numbers k such that x^6 == 1 (mod(k)) has no solution 1 < x < k-1.

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A172019 Numbers k such that 4 divides phi(k) (i.e., A000010(k)).

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

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A097987 Numbers k such that 4 does not divide phi(k), where phi is Euler's totient function (A000010).

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