OFFSET
1,2
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
FORMULA
Multiplicative with a(p^e) = (p^(e+2) - p)/(p^2 - 1) + 1 if e is odd and 1 otherwise.
a(n) >= 1, with equality if and only if n is a square (A000290).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^6/1080) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.51287686448947428073... .
MATHEMATICA
f[p_, e_] := If[OddQ[e], 1 + (p^(e + 2) - p)/(p^2 - 1), 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100]
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, if(f[i, 2]%2, (f[i, 1]^(f[i, 2]+2) - f[i, 1])/(f[i, 1]^2 - 1) + 1, 1)); }
CROSSREFS
KEYWORD
nonn,easy,mult
AUTHOR
Amiram Eldar, Dec 26 2023
STATUS
approved