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

-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

Author

N. J. A. Sloane, Sep 13 2003

Keywords

• sign
• easy
• nice

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]

Crossrefs

Cf. A006571 (all coefficients). A006962.

Programs

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 *)

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