OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p) = p^3 + p^2 + p + 1 for e <= 2, and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 3.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-3) + 1/p^(s-2) - 1/p^(2*s-4) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(4*s-4)).
Sum_{k=1..n} a(k) ~ c * n^4 / 4, where c = zeta(3) * zeta(4) * Product_{p prime} (1 - 1/p^3 - 1/p^4 + 1/p^7 + 1/p^12 - 1/p^13) = 1.00015013207437782094... .
MATHEMATICA
f[p_, e_] := (p^If[e <= 2, 4, e + 1]-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 50]
PROG
(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] <= 2, f[i, 2] = 3)); sigma(f); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 30 2024
STATUS
approved