A004124 -id:A004124 - cp's OEIS frontend

A193679 Sequence related to discriminant of cyclotomic polynomials A004124.

1, 2, 3, 4, 5, 12, 7, 16, 27, 80, 11, 144, 13, 448, 2025, 256, 17, 1728, 19, 6400, 35721, 11264, 23, 20736, 3125, 53248, 19683, 200704, 29, 518400, 31, 65536, 7144929, 1114112, 37515625, 2985984, 37, 4980736, 89813529, 40960000, 41, 146313216, 43, 126877696

Offset: 1

Wolfdieter Lang, Aug 20 2011

nonn

a(p) = p for primes p.

			n=6: a(6) = 2^(2/(2-1))*3^(2/(3-1)) = 12.
     Discriminant(Phi(6,x)) = -3 = - (6^phi(6))/a(6).

P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, p. 297, eq.(1).

Alois P. Heinz, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.
Eric Weisstein's World of Mathematics, Cyclotomic Polynomial

Cf. A004124.

with(numtheory): A193679 := n -> n^phi(n)/abs(discrim(cyclotomic (n,x),x)); seq(A193679(i),i=1..49); # Peter Luschny, Aug 20 2011

a[n_] := n^EulerPhi[n]/Abs[Discriminant[Cyclotomic[n, x], x]]; Array[a, 44]
(* Jean-François Alcover, Mar 21 2017 *)
Table[Product[d^(-n*MoebiusMu[d]/d), {d, Divisors[n]}], {n, 1, 50}] (* Vaclav Kotesovec, May 12 2024 *)
Table[Product[p^(EulerPhi[n]/(p-1)), {p, Select[Divisors[n], PrimeQ[#]&]}], {n, 1, 50}] (* Vaclav Kotesovec, May 13 2024 *)

a(n) = n^eulerphi(n)/abs(poldisc(polcyclo(n))); \\ Michel Marcus, Jul 14 2018

a(n) = n^phi(n)/abs(discriminant(Phi(n,x))), n>=1, with the cyclotomic polynomials Phi(n,x) and the Euler totient function phi(n)=A000010(n).
a(n) = product(p^(phi(n)/(p-1)),p prime dividing n), n>=2, a(1)=1.
Conjecture: Dirichlet g.f. of log(a(n)): -zeta(s-1)*zeta'(s)/zeta(s)^2, where zeta'(s) is the derivative of zeta(s). This would give a(n) = exp(Sum_{d|n} Lambda(d)*phi(n/d)), with Lambda(n)=log(A014963) and phi(n)=A000010. - Benedict W. J. Irwin, Jul 14 2018

A033949 Positive integers that do not have a primitive root.

Original entry on oeis.org

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.

A062981 a(n) = n^phi(n).

Original entry on oeis.org

1, 2, 9, 16, 625, 36, 117649, 4096, 531441, 10000, 25937424601, 20736, 23298085122481, 7529536, 2562890625, 4294967296, 48661191875666868481, 34012224, 104127350297911241532841, 25600000000, 7355827511386641

Offset: 1

Reinhard Zumkeller, Jul 24 2001

T. D. Noe, Table of n, a(n) for n = 1..100
Brett A. Harrison, On the reducibility of cyclotomic polynomials over finite fields, Am. Math. Monthly, Vol. 114, No. 9 (2007), pp. 813-818.

Cf. A000010, A004124, A193679, A239725.

[n^EulerPhi(n): n in [1..25]]; // Vincenzo Librandi, Apr 05 2017

A062981 := proc(n)
        n^numtheory[phi](n) ;
end proc:
seq(A062981(n),n=1..20) ; # R. J. Mathar, Oct 15 2011

Table[n^EulerPhi[n],{n,30}] (* Harvey P. Dale, Mar 30 2012 *)

a(n) = n^eulerphi(n); \\ Harry J. Smith, Aug 14 2009

a(n) = A001783(n) / (Product_{d|n} (d!/d^d)^A008683(n/d)) = (Product_{GCD(k, n)=1} k) / (Product_{d|n} (d!/d^d)^A008683(n/d)) = A175504(n) * n. - Jaroslav Krizek, May 31 2010

Sum_{n>=1} 1/a(n) = A239725. - Amiram Eldar, Nov 19 2020

A174311 Value of the n-th cyclotomic polynomial at the discriminant of that polynomial.

Original entry on oeis.org

0, 2, 7, 17, 246109501, 13, 22537999301860113141522943, 4294967297, 58149737003032434092905183, 242203001, 5313022609595033985218523349395070147785700752531778166637386100465086995951866123901089470951

Offset: 1

Artur Jasinski, Mar 15 2010

nonn

			C_4(x) = x^2 + 1 has discriminant -4 so a(4) = C_4(-4) = 17. - _Robert Israel_, Jul 19 2016

Robert Israel, Table of n, a(n) for n = 1..28

Cf. A004124.

Cf. A174304, A174305, A174306, A174307, A174308, A174309, A174310.

seq(numtheory:-cyclotomic(n, discrim(numtheory:-cyclotomic(n,x),x)), n=1..20); # Robert Israel, Jul 19 2016

s = {}; Do[d = Discriminant[Cyclotomic[n, x], x]; AppendTo[s, Cyclotomic[n, d]], {n, 1, 20}]; s

a(n) = C_n(A004124(n)) where A004124(n) is the discriminant of C_n(x) and C_n is the n-th cyclotomic polynomial. - Robert Israel, Jul 19 2016

Edited by Robert Israel, Jul 19 2016

A086783 Discriminant of the polynomial x^n - 1.

Original entry on oeis.org

1, 4, -27, -256, 3125, 46656, -823543, -16777216, 387420489, 10000000000, -285311670611, -8916100448256, 302875106592253, 11112006825558016, -437893890380859375, -18446744073709551616, 827240261886336764177, 39346408075296537575424, -1978419655660313589123979

Offset: 1

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

By definition, a(n) = Product_{1<=iJianing Song, Mar 17 2021

Kenneth G. Hawes, Table of n, a(n) for n = 1..386
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 249-255. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A000312, A004124.

a(n)=poldisc('x^n-1); \\ Joerg Arndt, May 04 2013

def A086783(n) : return (-1)^((n-1)//2) * n^n # Eric M. Schmidt, May 04 2013

a(n) = (-1)^floor((n-1)/2) * n^n = (-1)^floor((n-1)/2) * A000312(n).

More terms from Eric M. Schmidt, May 04 2013

A130614 a(n) = p^(p-2), where p = prime(n).

Original entry on oeis.org

1, 3, 125, 16807, 2357947691, 1792160394037, 2862423051509815793, 5480386857784802185939, 39471584120695485887249589623, 3053134545970524535745336759489912159909

Offset: 1

Jonathan Vos Post, Jun 18 2007

Number of labeled trees on p(n) nodes, where p(n) is the n-th prime.

Let p = prime(n). For n >= 2, (-1)^((p-1)/2) * a(n) is the discriminant of the p-th cyclotomic polynomial. - Jianing Song, May 10 2021

Vincenzo Librandi, Table of n, a(n) for n = 1..77

Cf. A000040, A000272, A004124, A036878.

Magma
```
[n^(n-2) : n in [2..40] | IsPrime(n)];
```

[p^(p-2): p in PrimesUpTo(50)]; // Vincenzo Librandi, Mar 27 2014

Table[Prime@n^(Prime@n - 2), {n, 20}] (* Vincenzo Librandi, Mar 27 2014 *)
#^(#-2)&/@Prime[Range[10]] (* Harvey P. Dale, Oct 18 2016 *)

a(n) = my(p=prime(n)); p^(p-2) \\ Felix Fröhlich, May 10 2021

a(n) = A000272(A000040(n)).

For n >= 2, (-1)^((p-1)/2) * a(n) = A004124(p), where p = prime(n). - Jianing Song, May 10 2021

Name edited by Felix Fröhlich, May 10 2021

A180634 Numbers n such that the discriminant of the n-th cyclotomic polynomial is a square.

Original entry on oeis.org

1, 2, 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

Offset: 1

Jan Fricke, Sep 13 2010

A number n is in this sequence if the Galois group of the n-th cyclotomic polynomial over the rationals contains only even permutations.
Essentially the same as A033949. - R. J. Mathar, Oct 15 2011
Also, numbers n such that the product of the elements in the group Z_n of invertible elements mod n (i.e., the product mod n of x such that 1 <= x < n and x is coprime to n) is 1. An equivalent characterization of the latter (apart from n=2): n such that the number of square roots of 1 mod n is divisible by 4. (See comments at A033949). - Robert Israel, Dec 08 2014
To see this, use Gauss's generalization of Wilson's theorem namely, the product of the units of Z_n is -1 if n is 4 or p^i or 2p^i for odd primes p, i >0,  and is equal to 1 otherwise. - W. Edwin Clark, Dec 09 2014

			n=5: The 5th cyclotomic polynomial is x^4+x^3+x^2+x+1 with discriminant 125, which is not a square. The Galois group is generated by (1243), that is an odd permutation. Hence 5 is not in the sequence. n=8: The 8th cyclotomic polynomial is x^4+1 with discriminant 256, which is a square. The Galois group is {id,(13)(57),(15)(37),(17)(35)}, that are all even permutations. Hence 8 is in the sequence.

Robert G. Wilson v, Table of n, a(n) for n = 1..1000
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A004124, A033949.

m := proc(n) local k, r; r := 1;
for k from 1 to n do if igcd(n,k) = 1 then r := modp(r*k,n) fi od; r end:
[1, op(select(n -> m(n) = 1, [$1..100]))]; # Peter Luschny, May 25 2017

fQ[n_] := IntegerQ@ Sqrt@ Discriminant[ Cyclotomic[ n, x], x]; Select[ Range@ 100, fQ] (* Robert G. Wilson v, Dec 10 2014 *)

for(n=1,100,if(issquare(poldisc(polcyclo(n))),print(n)))

A344407 Discriminant of the (2n)-th cyclotomic field Q(zeta_(2n)). Equivalently, discriminant of the (2n)-th cyclotomic polynomial.

Original entry on oeis.org

1, -4, -3, 256, 125, 144, -16807, 16777216, -19683, 4000000, -2357947691, 5308416, 1792160394037, 1157018619904, 1265625, 18446744073709551616, 2862423051509815793, 1586874322944, -5480386857784802185939, 1048576000000000000, 205924456521, 5829995856912430117421056

Offset: 1

Jianing Song, May 17 2021

Note that Q(zeta_n) = Q(zeta_(2n)) for odd n, so this sequence is A004124 with redundant values removed.

a(n) is negative <=> phi(2n) == 2 (mod 4) <=> n = 2 or n is of the form p^e, where p is a prime congruent to 3 modulo 4.

			n = 2: Q(zeta_4) = Q(i) has discriminant -4;
n = 3: Q(zeta_6) = Q(sqrt(-3)) has discriminant -3;
n = 4: Q(zeta_8) = Q(sqrt(2), i) has discriminant 256;
n = 5: Q(zeta_10) = Q(exp(2*Pi*i/5)) has discriminant 125;
n = 6: Q(zeta_12) = Q(sqrt(3), i) has discriminant 144.

Jianing Song, Table of n, a(n) for n = 1..200

Cf. A004124, A062570 (degree of Q(zeta_(2n))).

PARI
```
vector(25,n,poldisc(polcyclo(2*n)))
```

a(n) = A004124(2n). See A004124 for its formula.

For n >= 2, a(p) = (-1)^((p-1)/2) * A130614(n), where p = prime(n).

A215041 a(n) = n^degree(C(n,x))/discriminant(C(n,x)) for the minimal polynomials C(n,x) of 2*cos(Pi/n), given in A187360.

Original entry on oeis.org

1, 2, 3, 2, 5, 3, 7, 2, 9, 5, 11, 9, 13, 7, 45, 2, 17, 27, 19, 25, 189, 11, 23, 81, 125, 13, 243, 49, 29, 2025, 31, 2, 2673, 17, 6125, 729, 37, 19, 9477, 625, 41, 35721, 43, 121, 91125, 23, 47, 6561, 2401, 3125, 111537, 169, 53, 19683, 378125

Offset: 1

Wolfdieter Lang, Aug 24 2012

nonn

The discriminants for C(n,x), the minimal polynomial of 2*cos(Pi/n) are found under A193681. The degree of C(n,x), called delta(n), is given as A055034(n).

Compare this sequence with A193679, the anologon for the cyclotomic polynomials. See also the P. Ribenboim reference given in A004124.

			a(30) = 30^delta(30)/A193681(30) = 30^8/324000000 = 2025.
For the conjectures: i) a(4) = 2; ii) a^(3^2) = a(9) = 3^((3+1)/2) = 9; iii) a(30) = a(2*3*5) = 3^(delta(30)/2)*5^(delta(30)/4) = 3^4*5^2 = 2025;
  a(40) = a(2^3*5) = 5^(delta(40)/4) = 5^4 = 625; a(45) = a(3^2*5) = 3^(delta(45)/2)* 5^(delta(45)/4) = 91125.

Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics, Vol. 36, No. 2, 2007, pp. 251-257. Mathematical Reviews, MR2312537. Zentralblatt MATH, Zbl 1133.11012.

Cf. A193681, A055034, A193679 (cyclotomic case).

a(n) = (n^delta(n))/Discriminant(C(n,x)), n>=1, with the minimal polynomials C(n,x) of 2*cos(Pi/n), with coefficient triangle given in A187360, and their degree delta(n) given in A055034(n).
a(1) = 1. Conjectures for a(n), n>=2: i) a(2^k) = 2, k>=1;
  ii) a(p^k) = p^((p^(k-1)+1)/2), for odd prime p and k>=1;
  iii) a(n) = product(p^(delta(n)/(p-1)), odd p|n) otherwise.

A375745 a(n) is the sum of the vector of the reduced discriminant of the n-th cyclotomic polynomial.

Original entry on oeis.org

1, 1, 4, 4, 16, 4, 36, 16, 36, 16, 100, 16, 144, 36, 72, 64, 256, 36, 324, 64, 156, 100, 484, 64, 400, 144, 324, 144, 784, 72, 900, 256, 420, 256, 648, 144, 1296, 324, 600, 256, 1600, 156, 1764, 400, 648, 484, 2116, 256, 1764, 400, 1056, 576, 2704, 324, 1720, 576

Offset: 1

Darío Clavijo, Aug 26 2024

nonn

Conjecture: a(n) >= phi(n)^2 = A127473(n), and for n>=2 strictly greater iff n is in A070537.

Cf. A127473, A070537, A004124.

a(n) = vecsum(poldiscreduced(polcyclo(n)));

cp's OEIS Frontend

A193679 Sequence related to discriminant of cyclotomic polynomials A004124.

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A033949 Positive integers that do not have a primitive root.

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A062981 a(n) = n^phi(n).

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A174311 Value of the n-th cyclotomic polynomial at the discriminant of that polynomial.

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A086783 Discriminant of the polynomial x^n - 1.

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A130614 a(n) = p^(p-2), where p = prime(n).

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