A395773 The twisted Euler phi-function for the Dirichlet character kronecker(-15,.) mod 15.

1, 1, 3, 2, 5, 3, 8, 4, 9, 5, 12, 6, 14, 8, 15, 8, 16, 9, 18, 10, 24, 12, 22, 12, 25, 14, 27, 16, 30, 15, 30, 16, 36, 16, 40, 18, 38, 18, 42, 20, 42, 24, 44, 24, 45, 22, 46, 24, 56, 25, 48, 28, 52, 27, 60, 32, 54, 30, 60, 30, 60, 30, 72, 32, 70, 36, 68, 32, 66, 40

Offset

1,3

Comments

We have h(-15*n^2), the class number of the order of Z[(1+sqrt(-15))/2] with discriminant -15*n^2, is equal to 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(-15,.)) = 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(3^e) = 3^e, a(5^e) = 5^e, a(p^e) = p^(e-1)*(p-1) if p == 1, 2, 4, 8 (mod 15) and a(p^e) = p^(e-1)*(p+1) if p == 7, 11, 13, 14 (mod 15).

Example

Given L(0,kronecker(-15,.)) = 2, we have: h(-15*1^2) = 2*a(1) = 2, h(-15*2^2) = 2*a(2) = 2, h(-15*3^2) = 2*a(3) = 6, h(-15*4^2) = 2*a(4) = 4, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-15, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-15}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A316569 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, A395771, A395769, A395772, this sequence, A395774, A395775, A395776, A395777.

Sequence in context: A075889 A181771 A396715 * A238628 A045766 A281668

Adjacent sequences: A395770 A395771 A395772 * A395774 A395775 A395776

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved