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%I #13 Oct 25 2022 02:38:39
%S 1,3,5,11,7,15,9,31,18,21,13,55,15,27,35,75,19,54,21,77,45,39,25,155,
%T 38,45,58,99,31,105,33,167,65,57,63,198,39,63,75,217,43,135,45,143,
%U 126,75,49,375,66,114,95,165,55,174,91,279,105,93,61,385,63,99,162,355,105,195
%N a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).
%C Dirichlet convolution of [1,-1,0,4,0,0,...] with A007429.
%H Amiram Eldar, <a href="/A145398/b145398.txt">Table of n, a(n) for n = 1..10000</a>
%H J. S. Rutherford, <a href="https://dx.doi.org/10.1107/S0108767392000898">The enumeration and symmetry-significant properties of derivative lattices</a>, Act. Cryst. A48 (1992), 500-508. Table 1, symmetry C2/m.
%F Dirichlet g.f.: (1-1/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
%F From _Amiram Eldar_, Oct 25 2022: (Start):
%F Multiplicative with a(2^e) = 3*2^(e+1)-4*e-5, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
%F Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). (End)
%p read("transforms") ; s1 := [1,-1,0,4,seq(0,n=1..40)] ; s2 := [seq(add(sigma(d),d=divisors(n)),n=1..40)] ; DIRICHLET(s1,s2) ; # _R. J. Mathar_, Feb 07 2011
%t f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 3*2^(e + 1) - 4*e - 5; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*2^(f[i,2]+1) - 4*f[i,2] - 5, (f[i,1]*(f[i,1]^(f[i,2]+1)-1) - (f[i,1]-1)*(f[i,2]+1))/(f[i,1]-1)^2)); } \\ _Amiram Eldar_, Oct 25 2022
%Y Cf. A007429, A145378, A152649.
%K nonn,mult
%O 1,2
%A _N. J. A. Sloane_, Mar 13 2009
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