%I M0072 N0024 #55 Oct 09 2023 12:50:31
%S -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,
%T -16,18,10,9,8,-18,-7,10,-10,2,-7,4,-12,-6,-15,7,17,4,-2,0,12,19,18,
%U 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
%N Coefficient of x^p (p = n-th prime) in x * Product_{k>=1} (1-x^k)^2*(1-x^11k)^2.
%C 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, ...
%C 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
%D N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
%D N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
%H Seiichi Manyama, <a href="/A002070/b002070.txt">Table of n, a(n) for n = 1..10000</a> (first 1229 terms N. J. A. Sloane)
%H Goro Shimura, <a href="http://dx.doi.org/10.1515/crll.1966.221.209">A reciprocity law in non-solvable extensions</a>, J. Reine Angew. Math. 221 1966 209-220.
%H G. Shimura, <a href="/A002070/a002070.pdf">A reciprocity law in non-solvable extensions</a>, J. Reine Angew. Math. 221 1966 209-220. [Annotated scan of pages 218, 219 only]
%F a(n) == 1 + prime(n) (mod 5) if prime(n) != 11. - _Seiichi Manyama_, Sep 17 2016
%F 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
%t 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 *)
%Y Cf. A006571 (all coefficients). A006962.
%K sign,easy,nice
%O 1,1
%A _N. J. A. Sloane_, Sep 13 2003