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
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.
Crossrefs
Cf. A007429.
Programs
-
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
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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 *) -
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
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)
Comments