A109712 UnitarySigmaUnitaryPhi(n) or USUP(n).

1, 3, 2, 5, 4, 6, 6, 9, 8, 12, 10, 10, 12, 18, 8, 17, 16, 24, 18, 20, 12, 30, 22, 18, 24, 36, 26, 30, 28, 24, 30, 33, 20, 48, 24, 40, 36, 54, 24, 36, 40, 36, 42, 50, 32, 66, 46, 34, 48, 72, 32, 60, 52, 78, 40, 54, 36, 84, 58, 40, 60, 90, 48, 65, 48, 60, 66, 80, 44, 72, 70, 72, 72, 108, 48, 90, 60, 72, 78, 68

Offset

1,2

Comments

a(n) is defined as follows. If n = Product p_i^r_i then a(n) = UnitarySigma(2^r_1) *UnitaryPhi(n/2^r_1) = (2^r_1+1)*Product(p_i^r_i-1), 2<p_i. So a(n) = UnitarySigma(n) if n = 2^r, and a(n) = UnitaryPhi(n) if GCD(2,n) = 1.

Links

Amiram Eldar, Table of n, a(n) for n = 1..10000

Formula

a(n) = A034448(t)*A047994(n/t) where t = A006519(n).

Multiplicative with a(2^e) = 1+2^e, a(p^e) = p^e-1 for primes p>2, e>0. - R. J. Mathar, Jun 02 2011

Sum_{k=1..n} a(k) ~ c * n^2, where c = (7/10) * Product_{p prime} (1 - 1/(p*(p+1))) = (7/10) * A065463 = 0.493109... . - Amiram Eldar, Nov 17 2022

Example

a(2^4*7^2) = UnitarySigma(2^4) * UnitaryPhi(7^2) = 17*48 = 816.

Maple

A109712 := proc(n)
    local a ;
    a := 1;
    if n > 1 then
        for pe in ifactors(n)[2] do
            if op(1, pe) = 2 then
                a := a*(1+op(1, pe)^op(2, pe)) ;
            else
                a := a*(op(1, pe)^op(2, pe)-1) ;
            end if;
        end do:
    end if;
    a ;
end proc:
seq(A109712(n), n=1..100) ; # R. J. Mathar, Sep 04 2018

Mathematica

A034448[n_] := Sum[If[GCD[d, n/d] == 1, d, 0], {d, Divisors[n]}]; A047994[n_] := Times @@ (Power @@@ FactorInteger[n] - 1); A006519[n_] := 2^IntegerExponent[n, 2]; a[1] = 1; a[n_ /; IntegerQ[Log[2, n]]] := n+1; a[n_] := A034448[ A006519[n] ]*A047994[ n/A006519[n] ]; Table[a[n], {n, 1, 80}] (* Jean-François Alcover, Oct 03 2013 *)
f[p_, e_] := p^e - 1; f[2, e_] := 2^e + 1; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Nov 17 2022 *)

Crossrefs

Cf. A006519, A047994, A092760, A034448, A065463.

Sequence in context: A326730 A089279 A049820 * A095049 A118209 A109451

Adjacent sequences: A109709 A109710 A109711 * A109713 A109714 A109715

Keyword

nonn,easy,mult

Author

Yasutoshi Kohmoto, Aug 08 2005

Status

approved