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

Author

Yasutoshi Kohmoto, Aug 08 2005

Keywords

• nonn
• easy
• mult

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

Examples

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

Links

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

Crossrefs

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

Programs

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 *)

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

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