A395768 - OEIS

%I #17 May 07 2026 17:55:03

%S 1,2,4,4,4,8,8,8,12,8,12,16,12,16,16,16,16,24,20,16,32,24,24,32,20,24,

%T 36,32,28,32,32,32,48,32,32,48,36,40,48,32,40,64,44,48,48,48,48,64,56,

%U 40,64,48,52,72,48,64,80,56,60,64,60,64,96,64,48,96,68,64,96,64

%N The twisted Euler phi-function for the Dirichlet character kronecker(-4,.) mod 4.

%C For n > 1, we have h(-4*n^2), the class number of the order of Z[i] with discriminant -4*n^2, is equal to (1/2)*a(n).

%C In general, let d < 0 be a fundamental discriminant and n be a positive integer such that d*n^2 < -4, then h(d*n^2) = L(0,kronecker(d,.)) * n * Product_{p|n, p prime} (1 - kronecker(d,p)/p). This is a combination of Theorem 7.24 in "Primes of the form x^2+ny^2" by Cox and the Dirichlet class number formula. Here L(0,kronecker(-4,.)) = 1/2.

%D D. A. Cox, Primes of the form x^2+ny^2, Wiley, New York, 1989, p. 146.

%H Jianing Song, <a href="/A395768/b395768.txt">Table of n, a(n) for n = 1..10000</a>

%F Multiplicative with a(2^e) = 2^e, a(p^e) = p^(e-1)*(p-1) if p == 1 (mod 4) and a(p^e) = p^(e-1)*(p+1) if p == 3 (mod 4).

%e Given L(0,kronecker(-4,.)) = 1/2, we have: h(-4*2^2) = (1/2)*a(2) = 1, h(-4*3^2) = (1/2)*a(3) = 2, h(-4*4^2) = (1/2)*a(4) = 2, ...

%t f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-4, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, May 07 2026 *)

%o (PARI) a(n,{d=-4}) = my(f=factor(n)[,1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d,f[i])/f[i])

%Y A101455 is the corresponding Dirichlet character.

%Y Cf. for fundamental discriminants -3..-24: A227128, this sequence, A395771, A395769, A395772, A395773, A395774, A395775, A395776, A395777.

%K nonn,easy,mult

%O 1,2

%A _Jianing Song_, May 05 2026