A152958 Alladi's third-order function phi_3(n).

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, 18, 28, 8, 30, 28, 20, 16, 24, 24, 36, 18, 24, 24, 40, 12, 42, 30, 32, 22, 46, 28, 48, 24, 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

Offset

1,3

Comments

Generalizes phi_1(n) = A000010(n) and phi_2(n) = A047994(n).

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

Cf. A000010, A047994.

Sequence in context: A345061 A061020 A206369 * A278963 A308085 A178970

Adjacent sequences: A152955 A152956 A152957 * A152959 A152960 A152961

Keyword

mult,nonn,look

Author

R. J. Mathar, Mar 25 2012

Status

approved