This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A069184 #38 May 15 2025 08:24:09
%S A069184 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,
%T A069184 42,40,40,30,72,32,33,48,54,48,65,38,60,56,54,42,96,44,60,78,72,48,68,
%U A069184 57,93,72,70,54,120,72,72,80,90,60,120,62,96,104,65,84,144,68,90,96
%N A069184 Sum of divisors d of n such that d or n/d is odd.
%C A069184 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
%H A069184 Antti Karttunen, <a href="/A069184/b069184.txt">Table of n, a(n) for n = 1..16384</a>
%H A069184 <a href="/index/Su#sums_of_divisors">Index entries for sequences related to sums of divisors</a>
%F A069184 Multiplicative with a(2^e) = 2^e+1 and a(p^e) = (p^(e+1)-1)/(p-1) for an odd prime p.
%F A069184 G.f.: Sum_{m>0} m*x^m*(1+x^m+x^(2*m)-x^(3*m))/(1-x^(4*m)).
%F A069184 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]
%F A069184 Sum_{k=1..n} a(k) ~ 7*Pi^2*n^2 / 96. - _Vaclav Kotesovec_, Feb 08 2019
%e A069184 UOSigma(2^4*7^2) = UnitarySigma(2^4)*sigma(7^2) = 17*57 = 969.
%p A069184 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
%t A069184 Table[ Sum[ d*Boole[ OddQ[d] || OddQ[n/d] ], {d, Divisors[n]}], {n, 1, 69}] (* _Jean-François Alcover_, Mar 26 2013 *)
%t A069184 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 *)
%o A069184 (PARI) a(n) = sumdiv(n, d, d*((d % 2) || ((n/d) % 2))); \\ _Michel Marcus_, Apr 10 2014
%o A069184 (PARI) a(n)=my(e=valuation(n,2)); sigma(n>>e) * if(e,2^e+1,1) \\ _Charles R Greathouse IV_, Apr 10 2014
%Y A069184 Cf. A069733, A107749, A092356.
%K A069184 mult,nonn
%O A069184 1,2
%A A069184 _Vladeta Jovovic_, Apr 10 2002
%E A069184 Edited by _N. J. A. Sloane_, Aug 29 2008 at the suggestion of _R. J. Mathar_