A145511 Dirichlet g.f.: (1-2/2^s+7/4^s)*zeta(s)^3.
1, 1, 3, 7, 3, 3, 3, 19, 6, 3, 3, 21, 3, 3, 9, 37, 3, 6, 3, 21, 9, 3, 3, 57, 6, 3, 10, 21, 3, 9, 3, 61, 9, 3, 9, 42, 3, 3, 9, 57, 3, 9, 3, 21, 18, 3, 3, 111, 6, 6, 9, 21, 3, 10, 9, 57, 9, 3, 3, 63, 3, 3, 18, 91, 9, 9, 3, 21, 9, 9, 3, 114, 3, 3, 18, 21, 9, 9, 3, 111, 15, 3, 3, 63, 9, 3, 9, 57, 3, 18, 9
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, Acta Cryst. A48 (1992), 500-508. See Table 1, Symmetry Fmmm.
Programs
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Maple
read("transforms") : nmax := 100 : L := [1,-2,0,7,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 - 1)*e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Oct 25 2022 *)
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PARI
up_to = 65537; t1 = direuler(p=2, up_to, 1/(1-X)^3); t3 = direuler(p=2, 2, 1-2*X^1+7*X^2, up_to); v145511 = dirmul(t1, t3); A145511(n) = v145511[n]; \\ Antti Karttunen, Sep 27 2018, after R. J. Mathar's PARI-code for A158327
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PARI
a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i,1] == 2, 3*(f[i,2]-1)*f[i,2]+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-1)*e+1 and a(p^e) = (e+1)*(e+2)/2 if p > 2.
Sum_{k=1..n} a(k) ~ (7/8)*n*log(n)^2 + c_1*n*log(n) + c_2*n, where c_1 = 21*gamma/4 - 5*log(2)/2 - 7/4 and c_2 = 7/4 + 21*gamma*(gamma-1)/4 - 15*gamma*log(2)/2 - 21*gamma_1/4 + 5*log(2)/2 + 3*log(2)^2, where gamma is Euler's constant (A001620) and gamma_1 is the 1st Stieltjes constant (A082633). (End)
Comments