A069184 Sum of divisors d of n such that d or n/d is odd.

1, 3, 4, 5, 6, 12, 8, 9, 13, 18, 12, 20, 14, 24, 24, 17, 18, 39, 20, 30, 32, 36, 24, 36, 31, 42, 40, 40, 30, 72, 32, 33, 48, 54, 48, 65, 38, 60, 56, 54, 42, 96, 44, 60, 78, 72, 48, 68, 57, 93, 72, 70, 54, 120, 72, 72, 80, 90, 60, 120, 62, 96, 104, 65, 84, 144, 68, 90, 96

Offset: 1

Author

Vladeta Jovovic, Apr 10 2002

Keywords

• mult
• nonn

Comments

Might be called UnitaryOrdinarySigma(n): If n=Product p_i^r_i then UOSigma(n)=UnitarySigma(2^r_1)*Sigma(n/2^r_1)=(2^r_1+1)*Product (p_i^(r_i+1)-1)/(p_i-1), p_i is not 2. - Yasutoshi Kohmoto, Jun 11 2005

Examples

			UOSigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969.
		

Links

• Antti Karttunen, Table of n, a(n) for n = 1..16384
• Index entries for sequences related to sums of divisors

Crossrefs

Cf. A069733, A107749, A092356.

Programs

Maple

A069184 := proc(n) local a,f,p,e; a := 1 ; for f in ifactors(n)[2] do p := op(1,f) ; e := op(2,f) ; if p = 2 then a := a*(2^e+1) ; else a := a*(p^(e+1)-1)/(p-1) ; end if; end do; a ; end proc: # R. J. Mathar, Jun 02 2011

Mathematica

Table[ Sum[ d*Boole[ OddQ[d] || OddQ[n/d] ], {d, Divisors[n]}], {n, 1, 69}] (* Jean-François Alcover, Mar 26 2013 *)
f[2, e_] := 2^e+1; f[p_, e_] := (p^(e+1)-1)/(p-1); a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* Amiram Eldar, Sep 15 2020 *)

PARI

a(n) = sumdiv(n, d, d*((d % 2) || ((n/d) % 2))); \\ Michel Marcus, Apr 10 2014

PARI

a(n)=my(e=valuation(n,2)); sigma(n>>e) * if(e,2^e+1,1) \\ Charles R Greathouse IV, Apr 10 2014

Formula

Multiplicative with a(2^e) = 2^e+1 and a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p.

G.f.: Sum_{m>0} m*x^m*(1+x^m+x^(2*m)-x^(3*m))/(1-x^(4*m)).

Dirichlet g.f.: zeta(s) *zeta(s-1) *(1-2^(1-2s)). - R. J. Mathar, Jun 02 2011 [simplified by Michael Shamos, May 14 2025]

Sum_{k=1..n} a(k) ~ 7*Pi^2*n^2 / 96. - Vaclav Kotesovec, Feb 08 2019

Extensions

Edited by N. J. A. Sloane, Aug 29 2008 at the suggestion of R. J. Mathar