A224339 - OEIS

%I #21 Oct 13 2022 06:37:56

%S 1,5,13,29,31,65,57,125,121,155,133,377,183,285,403,509,307,605,381,

%T 899,741,665,553,1625,781,915,1093,1653,871,2015,993,2045,1729,1535,

%U 1767,3509,1407,1905,2379,3875,1723,3705,1893,3857,3751,2765,2257,6617,2801,3905,3991,5307

%N Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2.

%C Multiplicative because A113184 is.

%C Logarithmic derivative of A224340.

%H Paul D. Hanna, <a href="/A224339/b224339.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (-1)^n * Sum_{d|n^2} (-1)^d * d.

%F a(n) = A113184(n^2).

%F a(n) = sigma(n^2) for odd n; a(n) = 4*sigma(n^2/2) - sigma(n^2) for even n. - _Andrew Howroyd_, Jul 28 2018

%F Multiplicative with a(p^e) = 2^(2*e+1)-3 if p=2, and (p^(2*e+1)-1)/(p-1) otherwise. - _Amiram Eldar_, Jul 01 2022

%F Sum_{k=1..n} a(k) ~ c * n^3, where c = (9*zeta(3))/(2*Pi^2) = 0.548072... . - _Amiram Eldar_, Oct 13 2022

%e L.g.f.: L(x) = x + 5*x^2/2 + 13*x^3/3 + 29*x^4/4 + 31*x^5/5 + 65*x^6/6 + 57*x^7/7 + 125*x^8/8 + 121*x^9/9 + 155*x^10/10 +...

%e where

%e exp(L(x)) = 1 + x + 3*x^2 + 7*x^3 + 16*x^4 + 30*x^5 + 64*x^6 + 120*x^7 + 236*x^8 + 434*x^9 + 805*x^10 +...+ A224340(n)*x^n +...

%t dif[n_]:=Module[{divs=Divisors[n^2],od,ev},od=Total[Select[divs,OddQ]];ev=Total[Select[divs,EvenQ]];Abs[od-ev]]; Array[dif,60] (* _Harvey P. Dale_, Jul 16 2015 *)

%t f[p_, e_] := If[p == 2, 2^(2*e + 1) - 3, (p^(2*e + 1) - 1)/(p - 1)]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Array[a, 100] (* _Amiram Eldar_, Jul 01 2022 *)

%o (PARI) {a(n)=if(n<1, 0, (-1)^n*sumdiv(n^2, d, (-1)^d*d))}

%o for(n=1,64,print1(a(n),", "))

%o (PARI) a(n) = if(n%2, sigma(n^2), 4*sigma(n^2/2) - sigma(n^2)) \\ _Andrew Howroyd_, Jul 28 2018

%Y Cf. A065764, A113184, A224340.

%K nonn,mult

%O 1,2

%A _Paul D. Hanna_, Apr 03 2013