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A145378 a(n) = Sum_{d|n} sigma(d) - 2*Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

%I A145378 #17 Oct 25 2022 02:40:15
%S A145378 1,2,5,7,7,10,9,20,18,14,13,35,15,18,35,49,19,36,21,49,45,26,25,100,
%T A145378 38,30,58,63,31,70,33,110,65,38,63,126,39,42,75,140,43,90,45,91,126,
%U A145378 50,49,245,66,76,95,105,55,116,91,180,105,62,61,245,63,66,162,235,105,130,69
%N A145378 a(n) = Sum_{d|n} sigma(d) - 2*Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).
%C A145378 Dirichlet convolution of [1,-2,0,4,0,0,0,...] with A007429.
%H A145378 Amiram Eldar, <a href="/A145378/b145378.txt">Table of n, a(n) for n = 1..10000</a>
%H A145378 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 g(n).
%F A145378 Dirichlet g.f.: (1-2/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
%F A145378 From _Amiram Eldar_, Oct 25 2022: (Start)
%F A145378 Multiplicative with a(2^e) = 2^(e+2) - 3*(e+1) and a(p^e) = (p*(p^(e+1)-1) - (p-1)*(e+1))/(p-1)^2 if p > 2.
%F A145378 Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). (End)
%p A145378 with(numtheory); g:=proc(n) local d,c,b,t0,t1,t2,t3;
%p A145378 t1:=divisors(n);
%p A145378 t0:=add( sigma(d), d in t1);
%p A145378 t2:=0; for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
%p A145378 t3:=0; for d in t1 do if d mod 4 = 0 then t3:=t3+sigma(d/4); fi; od:
%p A145378 t0-2*t2+4*t3; end;
%p A145378 [seq(g(n),n=1..100)];
%p A145378 # alternative
%p A145378 read("transforms") : nmax := 100 :
%p A145378 L27 := [seq(i,i=1..nmax) ];
%p A145378 L := [1,-2,0,4,seq(0,i=1..nmax)] ;
%p A145378 DIRICHLET(L27,L) :
%p A145378 MOBIUSi(%) :
%p A145378 MOBIUSi(%) ; # _R. J. Mathar_, Sep 25 2017
%t A145378 a[n_] := Sum[DivisorSigma[1, d] - 2 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2] + 4 Boole[Mod[d, 4] == 0] DivisorSigma[1, d/4], {d, Divisors[n]}];
%t A145378 Array[a, 100] (* _Jean-François Alcover_, Apr 04 2020 *)
%t A145378 f[p_, e_] := (p*(p^(e + 1) - 1) - (p - 1)*(e + 1))/(p - 1)^2; f[2, e_] := 2^(e + 2) - 3*(e + 1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Oct 25 2022 *)
%o A145378 (PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 2^(f[i,2]+2) - 3*(f[i,2]+1),  (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 A145378 Cf. A007429, A300707.
%K A145378 nonn,mult
%O A145378 1,2
%A A145378 _N. J. A. Sloane_, Mar 12 2009