OFFSET
1,2
COMMENTS
Dirichlet convolution of [1,-1,0,4,0,0,...] 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. Table 1, symmetry C2/m.
FORMULA
Dirichlet g.f.: (1-1/2^s+4/4^s)*(zeta(s))^2*zeta(s-1).
From Amiram Eldar, Oct 25 2022: (Start):
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.
Sum_{k=1..n} a(k) ~ c * n^2, where c = Pi^4/72 = 1.352904... (A152649). (End)
MAPLE
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
MATHEMATICA
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 *)
PROG
(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
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
N. J. A. Sloane, Mar 13 2009
STATUS
approved