OFFSET
1,3
COMMENTS
n and a(n) have the same prime factors, except when 2 divides n but 4 does not divide n, then n/2 and a(n) have the same prime factors.
a(n) is negative <=> phi(n) == 2 (mod 4) <=> n = 4 or n is of the form p^e or 2*p^e, where p is a prime congruent to 3 modulo 4. - Jianing Song, May 17 2021
REFERENCES
E. R. Berlekamp, Algebraic Coding Theory, McGraw-Hill, NY, 1968, p. 91.
D. Marcus, Number Fields. Springer-Verlag, 1977, p. 27.
P. Ribenboim, Classical Theory of Algebraic Numbers, Springer, 2001, pp. 118-9 and p. 297.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
LINKS
Gheorghe Coserea, Table of n, a(n) for n = 1..388 (first 100 terms from T. D. Noe)
Mohammad K. Azarian, On the Hyperfactorial Function, Hypertriangular Function, and the Discriminants of Certain Polynomials, International Journal of Pure and Applied Mathematics 36(2), 2007, pp. 251-257. MR2312537. Zbl 1133.11012.
J. Shallit, Letter to N. J. A. Sloane, Mar 25 1980.
Eric Weisstein's World of Mathematics, Polynomial Discriminant.
FORMULA
Sign(a(n)) = (-1)^(phi(n)*(phi(n)-1)/2). Magnitude: For prime p, a(p) = p^(p-2). For n = p^e, a prime power, a(n) = p^(((p-1)*e-1)*p^(e-1)). For n = Product_{i=1..k} p_i^e_i, a product of prime powers, a(n) = Product_{i=1..k} a(p_i^e_i)^phi(n/p_i^e_i).
a(n) = Sign(a(n))*(n^phi(n))/(Product_{p|n, p prime} p^(phi(n)/(p-1))). See the Ribenboim reference, p. 297, eq.(1), with the sign taken from the previous formula and n=2 included. - Wolfdieter Lang, Aug 03 2011
EXAMPLE
a(100) = 2^40 * 5^70.
a(100) = ((-1)^(40*39/2))*(100^40)/(2^(40/1)*5^(40/4)) = +2^40*5^70. - Wolfdieter Lang, Aug 03 2011
MATHEMATICA
PrimePowers[n_] := Module[{f, t}, f=FactorInteger[n]; t=Transpose[f]; First[t]^Last[t]]; app[pp_] := Module[{f, p, e}, f=FactorInteger[pp]; p=f[[1, 1]]; e=f[[1, 2]]; p^(((p-1)e-1) p^(e-1))]; SetAttributes[app, Listable]; a[n_] := Module[{pp, phi=EulerPhi[n]}, If[n==1, 1, pp=PrimePowers[n]; (-1)^(phi*(phi-1)/2) Times@@(app[pp]^EulerPhi[n/pp])]]; Table[a[n], {n, 24}]
a[n_] := Discriminant[ Cyclotomic[n, x], x]; Table[a[n], {n, 1, 24}] (* Jean-François Alcover, Dec 06 2011 *)
PROG
(PARI) a(n) = poldisc(polcyclo(n));
(PARI)
a(n) = {
my(f = factor(n), fsz = matsize(f)[1],
g = prod(k=1, fsz, f[k, 1]),
h = prod(k=1, fsz, f[k, 1]-1), phi = (n\g)*h,
r = prod(k=1, fsz, f[k, 1] ^ ((phi\(f[k, 1]-1)) * (f[k, 2]*(f[k, 1]-1)-1))));
return((1-2*((phi\2)%2)) * r);
};
vector(24, n, a(n)) \\ Gheorghe Coserea, Oct 31 2016
CROSSREFS
KEYWORD
sign,easy,nice
AUTHOR
EXTENSIONS
Edited by T. D. Noe, Sep 30 2003
STATUS
approved