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
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).
Programs
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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 *) -
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
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)
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