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

1, 2, 5, 7, 7, 10, 9, 20, 18, 14, 13, 35, 15, 18, 35, 49, 19, 36, 21, 49, 45, 26, 25, 100, 38, 30, 58, 63, 31, 70, 33, 110, 65, 38, 63, 126, 39, 42, 75, 140, 43, 90, 45, 91, 126, 50, 49, 245, 66, 76, 95, 105, 55, 116, 91, 180, 105, 62, 61, 245, 63, 66, 162, 235, 105, 130, 69

Offset

1,2

Comments

Dirichlet convolution of [1,-2,0,4,0,0,0,...] with 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 g(n).

Formula

Dirichlet g.f.: (1-2/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).

From Amiram Eldar, Oct 25 2022: (Start)

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.

Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/96 = 1.01467803... (A300707). (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:
t3:=0; for d in t1 do if d mod 4 = 0 then t3:=t3+sigma(d/4); fi; od:
t0-2*t2+4*t3; end;
[seq(g(n), n=1..100)];
# alternative
read("transforms") : nmax := 100 :
L27 := [seq(i, i=1..nmax) ];
L := [1, -2, 0, 4, seq(0, i=1..nmax)] ;
DIRICHLET(L27, L) :
MOBIUSi(%) :
MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017

Mathematica

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]}];
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_] := 2^(e + 2) - 3*(e + 1); 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, 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

Crossrefs

Cf. A007429, A300707.

Sequence in context: A226213 A199590 A096624 * A069887 A254340 A120303

Adjacent sequences: A145375 A145376 A145377 * A145379 A145380 A145381

Keyword

nonn,mult

Author

N. J. A. Sloane, Mar 12 2009

Status

approved