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%I #17 Oct 25 2022 02:35:52
%S 1,1,3,7,3,3,3,19,6,3,3,21,3,3,9,37,3,6,3,21,9,3,3,57,6,3,10,21,3,9,3,
%T 61,9,3,9,42,3,3,9,57,3,9,3,21,18,3,3,111,6,6,9,21,3,10,9,57,9,3,3,63,
%U 3,3,18,91,9,9,3,21,9,9,3,114,3,3,18,21,9,9,3,111,15,3,3,63,9,3,9,57,3,18,9
%N Dirichlet g.f.: (1-2/2^s+7/4^s)*zeta(s)^3.
%C Dirichlet convolution of [1,-2,0,7,0,0,0,0,...] and A007425. - R. J. Mathar, Feb 07 2011
%H Antti Karttunen, <a href="/A145511/b145511.txt">Table of n, a(n) for n = 1..65537</a>
%H J. S. Rutherford, <a href="http://dx.doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Acta Cryst. A48 (1992), 500-508. See Table 1, Symmetry Fmmm.
%F From _Amiram Eldar_, Oct 25 2022: (Start)
%F Multiplicative with a(2^e) = 3*(e-1)*e+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.
%F Sum_{k=1..n} a(k) ~ (7/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 21*gamma/4 - 5*log(2)/2 - 7/4 and c_2 = 7/4 + 21*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 21*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)
%p read("transforms") :
%p nmax := 100 :
%p L := [1,-2,0,7,seq(0,i=1..nmax)] :
%p MOBIUSi(%) :
%p MOBIUSi(%) :
%p MOBIUSi(%) ; # _R. J. Mathar_, Sep 25 2017
%t f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := 3*(e - 1)*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o (PARI)
%o up_to = 65537;
%o t1 = direuler(p=2, up_to, 1/(1-X)^3);
%o t3 = direuler(p=2, 2, 1-2*X^1+7*X^2, up_to);
%o v145511 = dirmul(t1, t3);
%o A145511(n) = v145511[n]; \\ _Antti Karttunen_, Sep 27 2018, after _R. J. Mathar_'s PARI-code for A158327
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*(f[i,2]-1)*f[i,2]+1, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ _Amiram Eldar_, Oct 25 2022
%Y Cf. A007425, A145399, A145444, A145501, A158327, A158360.
%Y Cf. A001620, A082633.
%K nonn,mult
%O 1,3
%A _N. J. A. Sloane_, Mar 14 2009
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