OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p) = p^2 + p + 1 and a(p^e) = (p^(e+1)-1)/(p-1) for e >= 2.
Dirichlet g.f.: zeta(s-1) * zeta(s) * Product_{p prime} (1 + 1/p^(s-2) - 1/p^(2*s-3) - 1/p^(2*s-2) + 1/p^(3*s-3)).
Sum_{k=1..n} a(k) ~ c * n^3 / 3, where c = zeta(2) * zeta(3) * Product_{p prime} (1 - 1/p^2 - 1/p^3 + 1/p^5 + 1/p^6 - 1/p^7) = 1.01304866467771286896... .
MATHEMATICA
f[p_, e_] := If[e == 1, p^2 + p + 1, (p^(e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); for(i = 1, #f~, if(f[i, 2] == 1, f[i, 2] = 2)); sigma(f); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Jan 30 2024
STATUS
approved