A395769 The twisted Euler phi-function for the Dirichlet character kronecker(-8,.) mod 8.

1, 2, 2, 4, 6, 4, 8, 8, 6, 12, 10, 8, 14, 16, 12, 16, 16, 12, 18, 24, 16, 20, 24, 16, 30, 28, 18, 32, 30, 24, 32, 32, 20, 32, 48, 24, 38, 36, 28, 48, 40, 32, 42, 40, 36, 48, 48, 32, 56, 60, 32, 56, 54, 36, 60, 64, 36, 60, 58, 48, 62, 64, 48, 64, 84, 40, 66, 64, 48, 96

Offset

1,2

Comments

We have h(-8*n^2), the class number of the order of Z[sqrt(-2)] with discriminant -8*n^2, is equal to 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(-8,.)) = 1.

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, 3 (mod 8) and a(p^e) = p^(e-1)*(p+1) if p == 3, 5 (mod 8).

Example

Given L(0,kronecker(-8,.)) = 1, we have: h(-8*1^2) = a(1) = 1, h(-8*2^2) = a(2) = 2, h(-8*3^2) = a(3) = 2, h(-8*4^2) = a(4) = 4, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-8, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-8}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A188510 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, A395771, this sequence, A395772, A395773, A395774, A395775, A395776, A395777.

Sequence in context: A220358 A089002 A097089 * A317764 A318075 A318343

Adjacent sequences: A395766 A395767 A395768 * A395770 A395771 A395772

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved