OFFSET
1,2
COMMENTS
The number of these divisors is A365173(n).
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with 1 + p^e + (p^(e + 1) - p)/(p^2 - 1) if e is even, and 1 + p^e + (1 + p^(2*floor(e/4)+1))*(p^(2*floor((e+1)/4)+1) - p)/(p^2 - 1) if e is odd.
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(2)*zeta(6)/(2*zeta(3)) * Product_{p prime} (1 + 1/p^3 - 1/p^6) = 0.809912096042... .
MATHEMATICA
f[p_, e_] := 1 + p^e + If[EvenQ[e], (p^(e + 1) - p)/(p^2 - 1), (1 + p^(2*Floor[e/4] + 1))*(p^(2*Floor[(e + 1)/4] + 1) - p)/(p^2 - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n), p, e); prod(i = 1, #f~, p = f[i, 1]; e = f[i, 2]; 1 + p^e + if(e%2, (1 + p^(2*(e\4) + 1))*(p^(2*((e+1)\4) + 1) - p)/(p^2 - 1), (p^(e+1)-p)/(p^2-1))); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Aug 25 2023
STATUS
approved