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A109712 UnitarySigmaUnitaryPhi(n) or USUP(n). 4
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..80.

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

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-Fran├žois Alcover, Oct 03 2013 *)

CROSSREFS

Cf. A092760.

Sequence in context: A075491 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

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Last modified May 24 14:49 EDT 2019. Contains 323532 sequences. (Running on oeis4.)