A145399 Dirichlet g.f.: (1+4/2^s+1/4^s)*zeta(s)^3.
1, 7, 3, 19, 3, 21, 3, 37, 6, 21, 3, 57, 3, 21, 9, 61, 3, 42, 3, 57, 9, 21, 3, 111, 6, 21, 10, 57, 3, 63, 3, 91, 9, 21, 9, 114, 3, 21, 9, 111, 3, 63, 3, 57, 18, 21, 3, 183, 6, 42, 9, 57, 3, 70, 9, 111, 9, 21, 3, 171, 3, 21, 18, 127, 9, 63, 3, 57, 9, 63, 3, 222, 3, 21, 18, 57, 9, 63, 3, 183
Offset: 1
Links
- Antti Karttunen, Table of n, a(n) for n = 1..65537
- J. S. Rutherford, The enumeration and symmetry-significant properties of derivative lattices, Act. Cryst. A48 (1992), 500-508. Table 1, symmetry Pmmm.
Programs
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Maple
read("transforms") : nmax := 100 : L := [1,4,0,1,seq(0,i=1..nmax)] : MOBIUSi(%) : MOBIUSi(%) : MOBIUSi(%) ; # R. J. Mathar, Sep 25 2017 -
Mathematica
f[p_, e_] := (e + 1)*(e + 2)/2; f[2, e_] := 3*e*(e + 1) + 1;; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)
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PARI
t1=direuler(p=2,200,1/(1-X)^3) t2=direuler(p=2,2,1+4*X+X^2,200) t3=dirmul(t1,t2) A145399(n) = t3[n]; \\ This line added by Antti Karttunen, Sep 27 2018
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*f[i,2]*(f[i,2]+1)+1, (f[i,2]+1)*(f[i,2]+2)/2)); } \\ Amiram Eldar, Oct 25 2022
Formula
From Amiram Eldar, Oct 25 2022: (Start):
Multiplicative with a(2^e) = 3*e*(e+1)+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.
Sum_{k=1..n} a(k) ~ (13/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 39*gamma/4 - 5*log(2)/2 - 13/4 and c_2 = 13/4 + 39*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 39*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2/2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)