A277915 -id:A277915 - cp's OEIS frontend

A033949 Positive integers that do not have a primitive root.

8, 12, 15, 16, 20, 21, 24, 28, 30, 32, 33, 35, 36, 39, 40, 42, 44, 45, 48, 51, 52, 55, 56, 57, 60, 63, 64, 65, 66, 68, 69, 70, 72, 75, 76, 77, 78, 80, 84, 85, 87, 88, 90, 91, 92, 93, 95, 96, 99, 100, 102, 104, 105, 108, 110, 111, 112, 114, 115, 116, 117, 119, 120, 123

Offset: 1

Calculated by Jud McCranie

nonn

Numbers k such that the cyclotomic polynomial Phi(k,x) is reducible over Zp for all primes p. Harrison shows that this is equivalent to k > 2 and the discriminant of Phi(k,x), A004124(k), being a square. - T. D. Noe, Nov 06 2007
The multiplicative group modulo k is non-cyclic; the complement A033948. - Wolfdieter Lang, Mar 14 2012. See A281854 for the groups. - Wolfdieter Lang, Feb 04 2017
Numbers k with the property that there exists a positive integer m with 1 < m < k-1 and m^2 == 1 (mod k). - Reinhard Muehlfeld, May 27 2014
Also, numbers k for which A000010(k) > A002322(k), or equivalently A034380(k) > 1. - Ivan Neretin, Mar 28 2015
Numbers k of the form a + b + 2*sqrt(a*b + 1) for positive integers a,b such that a*b + 1 is a square. Proof: If 1 < m < k - 1 and m^2 == 1 (mod k), take a = (m^2 - 1)/k and b = ((k - m)^2 - 1)/k. Conversely, if k = a + b + 2*sqrt(a*b + 1), take m = a + sqrt(a*b + 1). - Tor Gunston, Apr 24 2021
Seems to be A050275 without the duplicates. - Charles R Greathouse IV, Feb 09 2025

I. Niven and H. S. Zuckerman, An Introduction to the Theory of Numbers, 4th edition, page 62, Theorem 2.25.

T. D. Noe, Table of n, a(n) for n = 1..10000
Brett A. Harrison, On the reducibility of cyclotomic polynomials over finite fields, Amer. Math. Monthly, Vol 114, No. 9 (2007), 813-818.
Eldar Sultanow, Christian Koch, and Sean Cox, Collatz Sequences in the Light of Graph Theory, Universität Potsdam (Germany, 2020).
Wikipedia, Primitive root modulo n

Cf. A000010, A002322, A033948 (complement), A193305 (composites with primitive root).

Column k=1 of A277915, A281854.

a033949 n = a033949_list !! (n-1)
a033949_list = filter
               (\x -> any ((== 1) . (`mod` x) . (^ 2)) [2 .. x-2]) [1..]
-- Reinhard Zumkeller, Dec 10 2014

m := proc(n) local k, r; r := 1; if n = 2 then return false fi;
for k from 1 to n do if igcd(n,k) = 1 then r := modp(r*k,n) fi od; r end:
select(n -> m(n) = 1, [$1..123]); # Peter Luschny, May 25 2017

Select[Range[2,130],!IntegerQ[PrimitiveRoot[#]]&] (* Harvey P. Dale, Oct 25 2011 *)
a[n_] := Module[{j, l = {}}, While[Length[l] CarmichaelLambda[j], AppendTo[l, j]; Break[]]]]; l[[n]]]; Array[a, 100] (* Jean-François Alcover, May 29 2018, after Alois P. Heinz's Maple code for A277915 *)

is(n)=n>7 && (!isprimepower(if(n%2,n,n/2)) || n>>valuation(n,2)==1) \\ Charles R Greathouse IV, Oct 08 2016

from itertools import count, islice
from sympy.ntheory import sqrt_mod_iter
def A033949_gen(): # generator of terms
    return filter(lambda n:max(filter(lambda k:k 1,count(3))
A033949_list = list(islice(A033949_gen(),30)) # Chai Wah Wu, Oct 26 2022

from sympy import primepi, integer_nthroot
def A033949(n):
    def f(x): return int(n+1+(x>=2)+(x>=4)+sum(primepi(integer_nthroot(x,k)[0])-1 for k in range(1,x.bit_length()))+sum(primepi(integer_nthroot(x>>1,k)[0])-1 for k in range(1,x.bit_length()-1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m # Chai Wah Wu, Feb 24 2025

[n for n in range(1,100) if not Integers(n).multiplicative_group_is_cyclic()]
# Ralf Stephan, Mar 30 2014

Positive integers except 1, 2, 4 and numbers of the form p^i and 2p^i, where p is an odd prime and i >= 1.

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

Original entry on oeis.org

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

cp's OEIS Frontend

A033949 Positive integers that do not have a primitive root.

Views

Author

Keywords

Comments

References

Links

Crossrefs

Programs

Haskell

Maple

Mathematica

PARI

Python

Python

Sage

Formula

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

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

Views

Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

PARI

Formula

Extensions

A305828 The n-th number m such that a nontrivial prime(n)-th root of unity modulo m exists.

Views

Author

Keywords

Comments

Links

Crossrefs

Formula