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A145396 a(n) = Sum_{d|n} sigma(d) + 3*Sum_{2c|n} sigma(c).

%I A145396 #16 Oct 25 2022 02:39:17
%S A145396 1,7,5,23,7,35,9,59,18,49,13,115,15,63,35,135,19,126,21,161,45,91,25,
%T A145396 295,38,105,58,207,31,245,33,291,65,133,63,414,39,147,75,413,43,315,
%U A145396 45,299,126,175,49,675,66,266,95,345,55,406,91,531,105,217,61,805,63,231,162,607
%N A145396 a(n) = Sum_{d|n} sigma(d) + 3*Sum_{2c|n} sigma(c).
%C A145396 Dirichlet convolution of [1,3,0,0,0,0,0,...] and A007429.
%H A145396 Amiram Eldar, <a href="/A145396/b145396.txt">Table of n, a(n) for n = 1..10000</a>
%H A145396 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. See Table 1, symmetry P2/m.
%F A145396 Dirichlet g.f.: (1+3/2^s)*zeta(s-1)*(zeta(s))^2.
%F A145396 From _Amiram Eldar_, Oct 25 2022: (Start):
%F A145396 Multiplicative with a(2^e) = 5*2^(e+1)-4*e-9, and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
%F A145396 Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*Pi^4/288 = 2.367582... . (End)
%p A145396 with(numtheory);
%p A145396 g:=proc(n)
%p A145396 local d,c,b,t0,t1,t2,t3;
%p A145396 t1:=divisors(n);
%p A145396 t0:=add( sigma(d), d in t1);
%p A145396 t2:=0;
%p A145396 for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
%p A145396 t0+3*t2;
%p A145396 end;
%p A145396 [seq(g(n),n=1..100)];
%p A145396 # alternative
%p A145396 nmax := 100 :
%p A145396 L27 := [seq(i,i=1..nmax) ];
%p A145396 L := [1,3,seq(0,i=1..nmax)] ;
%p A145396 MOBIUSi(%) ;
%p A145396 MOBIUSi(%) ;
%p A145396 DIRICHLET(%,L27) ; # _R. J. Mathar_, Sep 25 2017
%t A145396 a[n_] := Sum[DivisorSigma[1, d] + 3 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2], {d, Divisors[n]}];
%t A145396 Array[a, 100] (* _Jean-François Alcover_, Apr 04 2020 *)
%t A145396 f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 5*2^(e + 1) - 4*e - 9; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o A145396 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 5*2^(f[i,2]+1) - 4*f[i,2] - 9,  (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 A145396 Cf. A007429.
%K A145396 nonn,mult
%O A145396 1,2
%A A145396 _N. J. A. Sloane_, Mar 13 2009