|
%I #35 Nov 12 2022 05:25:08
%S 1,1,2,3,4,2,6,6,8,4,10,6,12,6,8,14,16,8,18,12,12,10,22,12,24,12,24,
%T 18,28,8,30,28,20,16,24,24,36,18,24,24,40,12,42,30,32,22,46,28,48,24,
%U 32,36,52,24,40,36,36,28,58,24,60,30,48,60,48,20,66,48,44,24,70,48,72,36,48,54,60,24
%N Alladi's third-order function phi_3(n).
%C Generalizes phi_1(n) = A000010(n) and phi_2(n) = A047994(n).
%H Amiram Eldar, <a href="/A152958/b152958.txt">Table of n, a(n) for n = 1..10000</a>
%H Krishnaswami Alladi, <a href="http://dx.doi.org/10.1017/S1446788700017304">On arithmetic functions and divisors of higher order</a>, J. Austral. Math. Soc. (Series A) 23 (1977) 9-27.
%F From _Amiram Eldar_, Nov 12 2022: (Start)
%F Multiplicative with a(p^e) = p^e * (1 - 1/p^ceiling((e+1)/2)).
%F 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)
%p Alladil := proc(y,star)
%p if star then
%p ceil(y) ;
%p else
%p if type(y,'integer') then
%p y+1 ;
%p else
%p ceil(y) ;
%p end if;
%p end if;
%p end proc:
%p Alladif := proc(r,x)
%p if r mod 2 = 1 then
%p Alladil( combinat[fibonacci](r-1)*x/combinat[fibonacci](r),false) ;
%p else
%p Alladil( combinat[fibonacci](r-1)*x/combinat[fibonacci](r),true) ;
%p end if;
%p end proc:
%p AlladiPhi := proc(r,n)
%p local pf,a,p,al;
%p a := 1 ;
%p for pf in ifactors(n)[2] do
%p p := op(1,pf) ;
%p al := op(2,pf) ;
%p a := a*(1-1/p^Alladif(r,al)) ;
%p end do;
%p return a*n ;
%p end proc: # implementation of theorem 3 of the Alladi paper
%p seq(AlladiPhi(3,n),n=1..80) ;
%t 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 *)
%o (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
%Y Cf. A000010, A047994.
%K mult,nonn,look
%O 1,3
%A _R. J. Mathar_, Mar 25 2012
|
