OFFSET
1,3
LINKS
Amiram Eldar, Table of n, a(n) for n = 1..10000
Krishnaswami Alladi, On arithmetic functions and divisors of higher order, J. Austral. Math. Soc. (Series A) 23 (1977) 9-27.
FORMULA
From Amiram Eldar, Nov 12 2022: (Start)
Multiplicative with a(p^e) = p^e * (1 - 1/p^ceiling((e+1)/2)).
Sum_{k=1..n} a(k) ~ c * n^2, where c = (zeta(3)/2) * Product_{p prime} (1 - 1/p^2 - 1/p^4 + 1/p^5) = 0.3462841864... . (End)
MAPLE
Alladil := proc(y, star)
if star then
ceil(y) ;
else
if type(y, 'integer') then
y+1 ;
else
ceil(y) ;
end if;
end if;
end proc:
Alladif := proc(r, x)
if r mod 2 = 1 then
Alladil( combinat[fibonacci](r-1)*x/combinat[fibonacci](r), false) ;
else
Alladil( combinat[fibonacci](r-1)*x/combinat[fibonacci](r), true) ;
end if;
end proc:
AlladiPhi := proc(r, n)
local pf, a, p, al;
a := 1 ;
for pf in ifactors(n)[2] do
p := op(1, pf) ;
al := op(2, pf) ;
a := a*(1-1/p^Alladif(r, al)) ;
end do;
return a*n ;
end proc: # implementation of theorem 3 of the Alladi paper
seq(AlladiPhi(3, n), n=1..80) ;
MATHEMATICA
f[p_, e_] := 1 - 1/p^Ceiling[(e+1)/2]; a[n_] := n * Times @@ (f @@@ FactorInteger[n]); Array[a, 100] (* Amiram Eldar, Aug 29 2019 *)
PROG
(PARI) a(n) = {my(f = factor(n)); n * prod(i = 1, #f~, 1 - 1/f[i, 1]^ceil((f[i, 2]+1)/2)); } \\ Amiram Eldar, Nov 12 2022
CROSSREFS
KEYWORD
AUTHOR
R. J. Mathar, Mar 25 2012
STATUS
approved