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A035179 a(n) = Sum_{d|n} Kronecker(-11, d). 7
1, 0, 2, 1, 2, 0, 0, 0, 3, 0, 1, 2, 0, 0, 4, 1, 0, 0, 0, 2, 0, 0, 2, 0, 3, 0, 4, 0, 0, 0, 2, 0, 2, 0, 0, 3, 2, 0, 0, 0, 0, 0, 0, 1, 6, 0, 2, 2, 1, 0, 0, 0, 2, 0, 2, 0, 0, 0, 2, 4, 0, 0, 0, 1, 0, 0, 2, 0, 4, 0, 2, 0, 0, 0, 6, 0, 0, 0, 0, 2, 5, 0, 0, 0, 0, 0, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,3

COMMENTS

This is a member of an infinite family of odd weight level 11 multiplicative modular forms. g_1 = A035179, g_3 = A129522, g_5 = A065099, g_7 = A138661. - Michael Somos, Jun 07 2015

Half of the number of integer solutions to x^2 + x*y + 3*y^2 = n. - Michael Somos, Jun 05 2005

From Jianing Song, Sep 07 2018: (Start)

Coefficients in expansion of Dirichlet series Product_p (1-(Kronecker(m,p)+1)*p^(-s) + Kronecker(m,p)*p^(-2s))^(-1) for m = -11.

Inverse Moebius transform of A011582. (End)

REFERENCES

H. McKean and V. Moll, Elliptic Curves, Cambridge University Press, 1997, page 202. MR1471703 (98g:14032)

LINKS

G. C. Greubel, Table of n, a(n) for n = 1..5000

FORMULA

a(n) is multiplicative with a(11^e) = 1, a(p^e) = (1 + (-1)^e) / 2 if p == 2, 6, 7, 8, 10 (mod 11), a(p^e) = e + 1 if p == 1, 3, 4, 5, 9 (mod 11). - Michael Somos, Jan 29 2007

Moebius transform is period 11 sequence [ 1, -1, 1, 1, 1, -1, -1, -1, 1, -1, 0, ...]. - Michael Somos, Jan 29 2007

G.f.: Sum_{k>0} Kronecker(-11, k) * x^k / (1 - x^k). - Michael Somos, Jan 29 2007

A028609(n) = 2 * a(n) unless n = 0. - Michael Somos, Jun 24 2011

EXAMPLE

G.f. = x + 2*x^3 + x^4 + 2*x^5 + 3*x^9 + x^11 + 2*x^12 + 4*x^15 + x^16 + 2*x^20 + ...

MATHEMATICA

a[ n_] := If[ n < 1, 0, DivisorSum[ n, KroneckerSymbol[ -11, #] &]]; (* Michael Somos, Jun 07 2015 *)

PROG

(PARI) {a(n) = if( n<1, 0, qfrep([2, 1; 1, 6], n, 1)[n])}; /* Michael Somos, Jun 05 2005 */

(PARI) {a(n) = if( n<1, 0, direuler(p=2, n, 1 / ((1 - X) * (1 - kronecker( -11, p)*X))) [n])}; /* Michael Somos, Jun 05 2005 */

(PARI) {a(n) = if( n<1, 0, sumdiv( n, d, kronecker( -11, d)))};

(MAGMA) A := Basis( ModularForms( Gamma1(11), 1), 88); B<q> := (-1 + A[1] + 2*A[2] + 4*A[4] + 2*A[5]) / 2; B; /* Michael Somos, Jun 07 2015 */

CROSSREFS

Cf. A028609, A065099, A129522, A138661.

Moebius transform gives A011582.

Sequence in context: A284688 A057595 A035201 * A035161 A035186 A035194

Adjacent sequences:  A035176 A035177 A035178 * A035180 A035181 A035182

KEYWORD

nonn,mult

AUTHOR

N. J. A. Sloane

STATUS

approved

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Last modified May 25 15:44 EDT 2019. Contains 323572 sequences. (Running on oeis4.)