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

1, 7, 5, 23, 7, 35, 9, 59, 18, 49, 13, 115, 15, 63, 35, 135, 19, 126, 21, 161, 45, 91, 25, 295, 38, 105, 58, 207, 31, 245, 33, 291, 65, 133, 63, 414, 39, 147, 75, 413, 43, 315, 45, 299, 126, 175, 49, 675, 66, 266, 95, 345, 55, 406, 91, 531, 105, 217, 61, 805, 63, 231, 162, 607

Offset

1,2

Comments

Dirichlet convolution of [1,3,0,0,0,0,0,...] and A007429.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. See Table 1, symmetry P2/m.

Formula

Dirichlet g.f.: (1+3/2^s)*zeta(s-1)*(zeta(s))^2.

From Amiram Eldar, Oct 25 2022: (Start):

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.

Sum_{k=1..n} a(k) ~ c * n^2, where c = 7*Pi^4/288 = 2.367582... . (End)

Maple

with(numtheory);
g:=proc(n)
local d, c, b, t0, t1, t2, t3;
t1:=divisors(n);
t0:=add( sigma(d), d in t1);
t2:=0;
for d in t1 do if d mod 2 = 0 then t2:=t2+sigma(d/2); fi; od:
t0+3*t2;
end;
[seq(g(n), n=1..100)];
# alternative
nmax := 100 :
L27 := [seq(i, i=1..nmax) ];
L := [1, 3, seq(0, i=1..nmax)] ;
MOBIUSi(%) ;
MOBIUSi(%) ;
DIRICHLET(%, L27) ; # R. J. Mathar, Sep 25 2017

Mathematica

a[n_] := Sum[DivisorSigma[1, d] + 3 Boole[Mod[d, 2] == 0] DivisorSigma[1, d/2], {d, Divisors[n]}];
Array[a, 100] (* Jean-François Alcover, Apr 04 2020 *)
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 *)

Prog

(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

Crossrefs

Cf. A007429.

Sequence in context: A166639 A078747 A213835 * A263825 A226661 A120404

Adjacent sequences: A145393 A145394 A145395 * A145397 A145398 A145399

Keyword

nonn,mult

Author

N. J. A. Sloane, Mar 13 2009

Status

approved