A107759 a(n) = (+2)UnitarySigma(n): if n = Product p_i^r_i then a(n) = Product (2 + p_i^r_i).

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, 60, 29, 54, 31, 140, 33, 34, 65, 76, 63, 66, 39, 84, 75, 70, 43, 180, 45, 78, 77, 100, 49, 90, 51, 108, 95, 90, 55, 116, 91

Offset: 1

Author

Yasutoshi Kohmoto, May 25 2005

Keywords

• nonn
• mult
• easy

Examples

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

Links

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

Crossrefs

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

Programs

Maple

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:
seq(A107759(n),n=1..60) ; # R. J. Mathar, Jan 07 2011

Mathematica

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

Formula

a(n) = Sum_{d|n, gcd(d, n/d) = 1} usigma(d), where usigma = A034448. - Ilya Gutkovskiy, Mar 27 2020

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