A107759 - OEIS

A107759

a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

%I #24 Nov 01 2022 07:13:53

%S 1,4,5,6,7,20,9,10,11,28,13,30,15,36,35,18,19,44,21,42,45,52,25,50,27,

%T 60,29,54,31,140,33,34,65,76,63,66,39,84,75,70,43,180,45,78,77,100,49,

%U 90,51,108,95,90,55,116,91

%N a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

%H Amiram Eldar, <a href="/A107759/b107759.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d), where usigma = A034448. - _Ilya Gutkovskiy_, Mar 27 2020

%F Sum_{k=1..n} a(k) ~ c * n^2, where c = (Pi^2/12) * Product_{p prime} (1 + 1/p^2 - 2/p^3) = A072691 * A330594 = 0.910438... . - _Amiram Eldar_, Nov 01 2022

%e a(12) = (2+3)*(2+4) = 30.

%p A107759 := proc(n) local pf,p ; if n = 1 then 1; else pf := ifactors(n)[2] ; mul( 2+op(1,p)^op(2,p), p=pf) ; end if; end proc:

%p seq(A107759(n),n=1..60) ; # _R. J. Mathar_, Jan 07 2011

%t a[1] = 1; a[n_] := Times @@ (2 + Power @@@ FactorInteger[n]); Array[a, 100] (* _Amiram Eldar_, Sep 20 2020 *)

%Y Cf. A007429, A034448, A054862, A072691, A107758, A330594.

%K nonn,mult,easy

%O 1,2

%A _Yasutoshi Kohmoto_, May 25 2005