OFFSET
1,2
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..1000
FORMULA
Multiplicative with a(p^e) = ((p-1)*(p+2))^e. If n = Product p(k)^e(k) then a(n) = Product ((p(k)-1)*(p(k)+2))^e(k).
Sum_{k>=1} 1/a(k) = Product_{primes p} (1 + 1/(p^2 + p - 3)) = 1.611922780552146990915794949248803526278171368254928942581015265238806543... - Vaclav Kotesovec, Sep 20 2020
Sum_{k=1..n} a(k) ~ c * n^3, where c = 2/(Pi^2 * Product_{p prime} (1 - 2/p^2 + 1/p^3 + 2/p^4)) = 0.3809790887... . - Amiram Eldar, Nov 05 2022
MATHEMATICA
a[1] = 1; a[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] - 1)^fi[[All, 2]])); b[1] = 1; b[n_] := (fi = FactorInteger[n]; Times @@ ((fi[[All, 1]] + 2)^fi[[All, 2]])); Table[a[n]*b[n], {n, 1, 100}] (* G. C. Greubel, Jun 10 2016 *)
f[p_, e_] := ((p - 1)*(p + 2))^e; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 05 2022 *)
PROG
(PARI) a(n) = {my(f = factor(n)); prod(i = 1, #f~, ((f[i, 1]-1)*(f[i, 1]+2))^f[i, 2]); } \\ Amiram Eldar, Nov 05 2022
CROSSREFS
KEYWORD
nonn,mult
AUTHOR
Jaroslav Krizek, Nov 01 2009
STATUS
approved