A395772 The twisted Euler phi-function for the Dirichlet character kronecker(-11,.) mod 11.

1, 3, 2, 6, 4, 6, 8, 12, 6, 12, 11, 12, 14, 24, 8, 24, 18, 18, 20, 24, 16, 33, 22, 24, 20, 42, 18, 48, 30, 24, 30, 48, 22, 54, 32, 36, 36, 60, 28, 48, 42, 48, 44, 66, 24, 66, 46, 48, 56, 60, 36, 84, 52, 54, 44, 96, 40, 90, 58, 48, 62, 90, 48, 96, 56, 66, 66, 108, 44, 96

Offset

1,2

Comments

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

Example

Given L(0,kronecker(-11,.)) = 1, we have: h(-11*1^2) = a(1) = 1, h(-11*2^2) = a(2) = 3, h(-11*3^2) = a(3) = 2, h(-11*4^2) = a(4) = 6, ...

Mathematica

f[p_, e_] := p^(e-1)*(p - KroneckerSymbol[-11, p]); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, May 07 2026 *)

Prog

(PARI) a(n, {d=-11}) = my(f=factor(n)[, 1]~, w=#f); n * prod(i=1, w, 1 - kronecker(d, f[i])/f[i])

Crossrefs

A011582 is the corresponding Dirichlet character.

Cf. for fundamental discriminants -3..-24: A227128, A395768, A395771, A395769, this sequence, A395773, A395774, A395775, A395776, A395777.

Sequence in context: A191444 A258241 A256739 * A267104 A093050 A289194

Adjacent sequences: A395769 A395770 A395771 * A395773 A395774 A395775

Keyword

nonn,easy,mult

Author

Jianing Song, May 05 2026

Status

approved