A193679 -id:A193679 - cp's OEIS frontend

A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

1, 1, 2, 3, 24, 5, 720, 105, 2240, 189, 3628800, 385, 479001600, 19305, 896896, 2027025, 20922789888000, 85085, 6402373705728000, 8729721, 47297536000, 1249937325, 1124000727777607680000, 37182145, 41363226782215962624, 608142583125, 1524503639859200000

Offset: 1

N. J. A. Sloane

In other words, a(1) = 1 and for n >= 2, a(n) = product of the phi(n) numbers < n and relatively prime to n.
From Gauss's generalization of Wilson's theorem (see Weisstein link) it follows that, for n>1, a(n) == -1 (mod n) if and only if there exists a primitive root modulo n (cf. the Hardy and Wright reference, Theorem 129. p. 102). - Vladimir Shevelev, May 11 2012
Islam & Manzoor prove that a(n) is an injection for n > 1, see links. In other words, if a(m) = a(n), and min(m, n) > 1, then m = n. - Muhammed Hedayet, May 25 2016
Cosgrave & Dilcher propose the name Gauss factorial. Indeed the sequence is the special case N = n of the Gauss factorial N_n! = Product_{1<=j<=N, gcd(j, n)=1} j (see A216919). - Peter Luschny, Feb 07 2018

G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers, Fifth ed., Clarendon Press, Oxford, 2003, Theorem 129, p. 102.
N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n = 1..200
J. B. Cosgrave and K. Dilcher, Extensions of the Gauss-Wilson Theorem, Integers: Electronic Journal of Combinatorial Number Theory, 8 (2008)
J. B. Cosgrave and K. Dilcher, The multiplicative orders of certain Gauss factorials, Intl. J. Number Theory 7 (1) (2011) 145-171.
S. W. Golomb and William Small, Problem E1045, Amer. Math. Monthly, 60 (1953), 422.
Muhammed H. Islam and Shahriar Manzoor, φ1 and phitorial are injections, for any positive integer N, where N > 1
Laszlo Toth, Weighted gcd-sum functions, J. Integer Sequences, 14 (2011), Article 11.7.7
Eric Weisstein's World of Mathematics, Wilson's Theorem

Cf. A023896, A066570, A038566, A124441.

Main diagonal gives A216919.

a001783 = product . a038566_row
-- Reinhard Zumkeller, Mar 04 2012, Aug 26 2011

A001783 := proc(n) local i,t1; t1 := 1; for i from 1 to n do if gcd(i,n)=1 then t1 := t1*i; fi; od; t1; end;
A001783 := proc(n) local i; mul(i,i=select(k->igcd(n,k)=1,[$1..n])) end; # Peter Luschny, Oct 30 2010

A001783[n_]:=Times@@Select[Range[n],CoprimeQ[n,#]&];
Array[A001783,20] (* Enrique Pérez Herrero, Jul 23 2011 *)

A001783(n)=prod(k=2,n-1,k^(gcd(k,n)==1))  \\ M. F. Hasler, Jul 23 2011

a(n)=my(f=factor(n),t=n^eulerphi(f)); fordiv(f,d, t*=(d!/d^d)^moebius(n/d)); t \\ Charles R Greathouse IV, Nov 05 2015

def Gauss_factorial(N, n): return mul(j for j in (1..N) if gcd(j, n) == 1)
def A001783(n): return Gauss_factorial(n, n)
[A001783(n) for n in (1..25)] # Peter Luschny, Oct 01 2012

a(n) = n^phi(n)*Product_{d|n} (d!/d^d)^mu(n/d); phi=A000010 is the Euler totient function and mu=A008683 the Moebius function (Tom M. Apostol, Introduction to Analytic Number Theory, New York 1984, p. 48). - Franz Vrabec, Jul 08 2005
a(n) = n!/A066570(n). - R. J. Mathar, Mar 10 2011
A001221(a(n)) = A000720(n) - A001221(n) = A048865(n).
A006530(a(n)) = A136548(n). - Enrique Pérez Herrero, Jul 23 2011
a(n) = A124441(n)*A124442(n). - M. F. Hasler, Jul 23 2011
a(n) == (-1)^A211487(n) (mod n). - Vladimir Shevelev, May 13 2012
a(n) = A250269(n) / A193679(n). - Daniel Suteu, Apr 05 2021

More terms from James Sellers, Dec 23 1999

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

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.

A372793 Sequence related to the asymptotic expansion of Sum_{k=1..n} tau(m*k).

Original entry on oeis.org

1, 2, 3, 16, 5, 864, 7, 4096, 729, 64000, 11, 6879707136, 13, 2809856, 61509375, 4294967296, 17, 812479653347328, 19, 26843545600000000, 26795786661, 2791309312, 23, 4019988717840603673710821376, 9765625, 73719087104, 7625597484987, 25962355635465062711296, 29

Offset: 1

Vaclav Kotesovec, May 13 2024

nonn

For m>=1, Sum_{k=1..n} tau(m*k) = A018804(m) * n * log(n) + O(n).

If p is prime, then Sum_{k=1..n} tau(p*k) ~ (2*p - 1) * n * (log(n) - 1 + 2*gamma)/p + n*log(p)/p, where gamma is the Euler-Mascheroni constant A001620.

			Sum_{k=1..n} tau(4*k) ~ (8*n*(log(n) + 2*gamma - 1) + n*4*log(2)) / 4, a(4) = exp(4*log(2)) = 16.
Sum_{k=1..n} tau(6*k) ~ (15*n*(log(n) + 2*gamma - 1) + n*(5*log(2) + 3*log(3))) / 6, a(6) = exp(5*log(2) + 3*log(3)) = 864.
Sum_{k=1..n} tau(8*k) ~ (20*n*(log(n) + 2*gamma - 1) + n*12*log(2)) / 8, a(8) = exp(12*log(2)) = 4096.
Sum_{k=1..n} tau(9*k) ~ (21*n*(log(n) + 2*gamma - 1) + n*6*log(3)) / 9, a(9) = exp(6*log(3)) = 729.
Sum_{k=1..n} tau(10*k) ~ (27*n*(log(n) + 2*gamma - 1) + n*(9*log(2) + 3*log(5))) / 10, a(10) = exp(9*log(2) + 3*log(5)) = 64000.
Sum_{k=1..n} tau(12*k) ~ (40*n*(log(n) + 2*gamma - 1) + n*(20*log(2) + 8*log(3))) / 12, a(12) = exp(20*log(2) + 8*log(3)) = 6879707136.

Cf. A000005 (m=1), A099777 (m=2), A372713 (m=3), A372784 (m=4), A372785 (m=5), A372786 (m=6), A372787 (m=7), A372788 (m=8), A372789 (m=9), A372790 (m=10), A372791 (m=11), A372792 (m=12).

Cf. A018804, A193679.

Sum_{k=1..n} tau(m*k) ~ A018804(m) * n * (log(n) - 1 + 2*gamma)/m + n*log(a(m))/m.
a(m) = exp(limit_{n->oo} (m * (Sum_{k=1..n} tau(m*k)) - A018804(m)*n*(log(n) - 1 + 2*gamma))/n).
If p is prime, then a(p) = p.
If p is prime, then a(p^k) = p^(k*p^(k-1)).
If p and q are distinct primes, then a(p*q) = p^(2*q-1) * q^(2*p-1).

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A001783 n-phi-torial, or phi-torial of n: Product k, 1 <= k <= n, k relatively prime to n.

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

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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.

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A372793 Sequence related to the asymptotic expansion of Sum_{k=1..n} tau(m*k).

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