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UnitarySigmaUnitaryPhi(n) or USUP(n).
5

%I #20 Nov 17 2022 05:15:21

%S 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,

%T 26,30,28,24,30,33,20,48,24,40,36,54,24,36,40,36,42,50,32,66,46,34,48,

%U 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

%N UnitarySigmaUnitaryPhi(n) or USUP(n).

%C 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.

%H Amiram Eldar, <a href="/A109712/b109712.txt">Table of n, a(n) for n = 1..10000</a>

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

%F 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

%F 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

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

%p A109712 := proc(n)

%p local a ;

%p a := 1;

%p if n > 1 then

%p for pe in ifactors(n)[2] do

%p if op(1,pe) = 2 then

%p a := a*(1+op(1,pe)^op(2,pe)) ;

%p else

%p a := a*(op(1,pe)^op(2,pe)-1) ;

%p end if;

%p end do:

%p end if;

%p a ;

%p end proc:

%p seq(A109712(n),n=1..100) ; # _R. J. Mathar_, Sep 04 2018

%t 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 *)

%t 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 *)

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

%K nonn,easy,mult

%O 1,2

%A _Yasutoshi Kohmoto_, Aug 08 2005