A145398 a(n) = Sum_{d|n} sigma(d) - Sum_{2c|n} sigma(c) + 4*Sum_{4b|n} sigma(b).

1, 3, 5, 11, 7, 15, 9, 31, 18, 21, 13, 55, 15, 27, 35, 75, 19, 54, 21, 77, 45, 39, 25, 155, 38, 45, 58, 99, 31, 105, 33, 167, 65, 57, 63, 198, 39, 63, 75, 217, 43, 135, 45, 143, 126, 75, 49, 375, 66, 114, 95, 165, 55, 174, 91, 279, 105, 93, 61, 385, 63, 99, 162, 355, 105, 195

Offset: 1

Author

N. J. A. Sloane, Mar 13 2009

Keywords

• nonn
• mult

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.

Crossrefs

Cf. A007429, A145378, A152649.

Programs

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 *)

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

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)