A002070 Coefficient of x^p (p = n-th prime) in x * Product_{k>=1} (1-x^k)^2*(1-x^11k)^2. (Formerly M0072 N0024)

-2, -1, 1, -2, 1, 4, -2, 0, -1, 0, 7, 3, -8, -6, 8, -6, 5, 12, -7, -3, 4, -10, -6, 15, -7, 2, -16, 18, 10, 9, 8, -18, -7, 10, -10, 2, -7, 4, -12, -6, -15, 7, 17, 4, -2, 0, 12, 19, 18, 15, 24, -30, -8, -23, -2, 14, 10, -28, -2, -18, 4, 24, 8, 12, -1, 13, 7, -22, 28, 30, -21, -20, -17, -26, -5, -1, -15, -2

Offset

1,1

Comments

Form the infinite product x*[(1-x)*(1-x^11)*(1-x^2)*(1-x^22)*(1-x^3)*(1-x^33)*(1-x^4)*(1-x^44)*...]^2 and take the coefficients of x^2, x^3, x^5, x^7, x^11, x^13, x^17, x^19, ...

The primes p where A006571(p) == 0 (mod p) are called supersingular for the elliptic curve "11a3" and are given by sequence A006962. - Michael Somos, Dec 25 2010

References

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).

N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

Links

Seiichi Manyama, Table of n, a(n) for n = 1..10000 (first 1229 terms N. J. A. Sloane)

Goro Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220.

G. Shimura, A reciprocity law in non-solvable extensions, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]

Formula

a(n) == 1 + prime(n) (mod 5) if prime(n) != 11. - Seiichi Manyama, Sep 17 2016

Conjecture: a(n) = Sum_{k=1..prime(n)} Sum_{y=1..prime(n)} Sum_{x=1..prime(n)} (A023900(k)/prime(n))[GCD(f(x,y), prime(n)) = k], where f(x,y) = x^3 - x^2 - y^2 - y. - Mats Granvik, Oct 09 2023

Mathematica

a[ n_] := If[ n < 1, 0, With[ {m = Prime @ n}, SeriesCoefficient[ q (Product[ (1 - q^(11 k)), {k, Ceiling[m/11]}]Product[ 1 - q^k, {k, m}])^2, {q, 0, m}]]] (* Michael Somos, Jul 04 2011 *)

Crossrefs

Cf. A006571 (all coefficients). A006962.

Sequence in context: A165585 A082506 A053000 * A326376 A106052 A050473

Adjacent sequences: A002067 A002068 A002069 * A002071 A002072 A002073

Keyword

sign,easy,nice

Author

N. J. A. Sloane, Sep 13 2003

Status

approved