A395768 The twisted Euler phi-function for the Dirichlet character kronecker(-4,.) mod 4.

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, 36, 32, 28, 32, 32, 32, 48, 32, 32, 48, 36, 40, 48, 32, 40, 64, 44, 48, 48, 48, 48, 64, 56, 40, 64, 48, 52, 72, 48, 64, 80, 56, 60, 64, 60, 64, 96, 64, 48, 96, 68, 64, 96, 64

Offset

1,2

Comments

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).

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.

References

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

Links

Jianing Song, Table of n, a(n) for n = 1..10000

Formula

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).

Example

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, ...

Mathematica

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 *)

Prog

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

Crossrefs

A101455 is the corresponding Dirichlet character.

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

Sequence in context: A117726 A172307 A108039 * A367013 A103228 A076340

Adjacent sequences: A395765 A395766 A395767 * A395769 A395770 A395771

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved