A395771 The twisted Euler phi-function for the Dirichlet character kronecker(-7,.) mod 7.

1, 1, 4, 2, 6, 4, 7, 4, 12, 6, 10, 8, 14, 7, 24, 8, 18, 12, 20, 12, 28, 10, 22, 16, 30, 14, 36, 14, 28, 24, 32, 16, 40, 18, 42, 24, 36, 20, 56, 24, 42, 28, 42, 20, 72, 22, 48, 32, 49, 30, 72, 28, 52, 36, 60, 28, 80, 28, 60, 48, 62, 32, 84, 32, 84, 40, 66, 36, 88, 42

Offset

1,3

Comments

We have h(-7*n^2), the class number of the order of Z[(1+sqrt(-7))/2] with discriminant -7*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(-7,.)) = 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(7^e) = 7^e, a(p^e) = p^(e-1)*(p-1) if p == 1, 2, 4 (mod 7) and a(p^e) = p^(e-1)*(p+1) if p == 3, 5, 6 (mod 7).

Example

Given L(0,kronecker(-7,.)) = 1, we have: h(-7*1^2) = a(1) = 1, h(-7*2^2) = a(2) = 1, h(-7*3^2) = a(3) = 4, h(-7*4^2) = a(4) = 2, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-7, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-7}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A175629 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, this sequence, A395769, A395772, A395773, A395774, A395775, A395776, A395777.

Sequence in context: A386713 A162339 A200697 * A253629 A327095 A176836

Adjacent sequences: A395768 A395769 A395770 * A395772 A395773 A395774

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved