A395774 The twisted Euler phi-function for the Dirichlet character kronecker(-19,.) mod 19.

1, 3, 4, 6, 4, 12, 6, 12, 12, 12, 10, 24, 14, 18, 16, 24, 16, 36, 19, 24, 24, 30, 22, 48, 20, 42, 36, 36, 30, 48, 32, 48, 40, 48, 24, 72, 38, 57, 56, 48, 42, 72, 42, 60, 48, 66, 46, 96, 42, 60, 64, 84, 54, 108, 40, 72, 76, 90, 60, 96, 60, 96, 72, 96, 56, 120, 68, 96, 88, 72

Offset

1,2

Comments

We have h(-19*n^2), the class number of the order of Z[(1+sqrt(-19))/2] with discriminant -19*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(-19,.)) = 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(19^e) = 19^e, a(p^e) = p^(e-1)*(p-1) if p == 1, 4, 5, 6, 7, 9, 11, 16, 17 (mod 19) and a(p^e) = p^(e-1)*(p+1) if p == 2, 3, 8, 10, 12, 13, 14, 15, 18 (mod 19).

Example

Given L(0,kronecker(-19,.)) = 1, we have: h(-19*1^2) = a(1) = 1, h(-19*2^2) = a(2) = 3, h(-19*3^2) = a(3) = 4, h(-19*4^2) = a(4) = 6, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-19, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-19}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A011585 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, A395771, A395769, A395772, A395773, this sequence, A395775, A395776, A395777.

Sequence in context: A279081 A139186 A326417 * A047840 A037189 A083342

Adjacent sequences: A395771 A395772 A395773 * A395775 A395776 A395777

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved