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%I #14 Sep 23 2018 00:21:37
%S 0,0,0,0,0,0,0,0,0,0,1,0,2,1,1,2,5,2,7,3,5,4,12,5,12,6,13,8,22,7,26,
%T 13,19,11,25,13,40,14,29,19,51,13,57,25,39,21,70,23,69,24,55,37,92,22,
%U 79,42,71,34,117,34,126,39,87,61,117,31,155,68,109,45,176,55,187,56,119,87
%N Dimension of the space of newforms of weight 2 on the subgroup Gamma_1(n).
%H G. C. Greubel, <a href="/A159046/b159046.txt">Table of n, a(n) for n = 1..10000</a>
%H G. Martin, <a href="https://dx.doi.org/10.1016/j.jnt.2004.10.009">Dimensions of the spaces of cusp forms and newforms on Gamma_0(N) and Gamma_1(N)</a>, J. Numb. Theory 112 (2005) 298-331.
%F a(n) = A029937(n) - sum a(m)*d(n/m), where the summation is over all divisors 1 < m < n of n and d is the divisor function.
%F Dirichlet convolution of A007247 and A029937. - _Michael Somos_, May 10 2015
%e a(p) = A029937(p) = (p-5)*(p-7)/24 for any prime p>3.
%e G.f. = x^11 + 2*x^13 + x^14 + x^15 + 2*x^16 + 5*x^17 + 2*x^18 + 7*x^19 + ...
%t a[ n_] := If[ n < 1, 0, Sum[ DivisorSum[ n/j, MoebiusMu[#] MoebiusMu[n/j/#] &] If[ j < 5, 0, 1 + DivisorSum[ j, #^2 MoebiusMu[ j/#] / 24 - EulerPhi [#] EulerPhi[j/#] / 4 &]], {j, Divisors@n}]]; (* _Michael Somos_, May 10 2015 *)
%o (PARI) {a(n) = if( n<1, 0, sumdiv(n, j, sumdiv(n/j, k, moebius(k) * moebius(n/j/k)) * if( j<5, 0, 1 + sumdiv(j, k, k^2 * moebius(j/k) / 24 - eulerphi(k) * eulerphi(j/k) / 4))))}; /* _Michael Somos_, May 10 2015 */
%Y Cf. A007427, A029937, A029938, A127788.
%K nonn
%O 1,13
%A _Steven Finch_, Apr 03 2009
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