A224340 -id:A224340 - cp's OEIS frontend

A225925 G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

1, 1, -2, 2, -1, -7, 8, -14, 1, 11, -23, 43, -54, 38, 17, -55, 162, -198, 257, -175, 69, 141, -518, 764, -1049, 1215, -1241, 549, 161, -1625, 3192, -5176, 6782, -7568, 7267, -4263, -788, 8394, -17866, 29782, -39041, 46101, -45857, 36551, -14591, -20937, 70638, -129520, 190994, -245846, 280560

Offset: 0

Paul D. Hanna, May 20 2013

sign

Compare to: Sum_{n>=0} x^(n*(n+1)/2) = exp( Sum_{n>=1} A002129(n)*x^n/n ).

			G.f.: A(x) = 1 + x - 2*x^2 + 2*x^3 - x^4 - 7*x^5 + 8*x^6 - 14*x^7 + x^8 +...
where
log(A(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 +...+ A002129(n^2)*x^n/n +...

Paul D. Hanna, Table of n, a(n) for n = 0..1000

Cf. A224340, A224339, A002129; variant: A215603.

{A002129(n)=if(n<1, 0, -sumdiv(n, d, (-1)^d*d))}
{a(n)=polcoeff(exp(sum(k=1,n,A002129(k^2)*x^k/k)+x*O(x^n)),n)}
for(n=0,50,print1(a(n),", "))

A224339 Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2.

Original entry on oeis.org

1, 5, 13, 29, 31, 65, 57, 125, 121, 155, 133, 377, 183, 285, 403, 509, 307, 605, 381, 899, 741, 665, 553, 1625, 781, 915, 1093, 1653, 871, 2015, 993, 2045, 1729, 1535, 1767, 3509, 1407, 1905, 2379, 3875, 1723, 3705, 1893, 3857, 3751, 2765, 2257, 6617, 2801, 3905, 3991, 5307

Offset: 1

Paul D. Hanna, Apr 03 2013

Multiplicative because A113184 is.

Logarithmic derivative of A224340.

			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 +...
where
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 +...

Paul D. Hanna, Table of n, a(n) for n = 1..1000

Cf. A065764, A113184, A224340.

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 *)
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 *)

{a(n)=if(n<1, 0, (-1)^n*sumdiv(n^2, d, (-1)^d*d))}
for(n=1,64,print1(a(n),", "))

a(n) = if(n%2, sigma(n^2), 4*sigma(n^2/2) - sigma(n^2)) \\ Andrew Howroyd, Jul 28 2018

a(n) = (-1)^n * Sum_{d|n^2} (-1)^d * d.
a(n) = A113184(n^2).
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
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
Sum_{k=1..n} a(k) ~ c * n^3, where c = (9*zeta(3))/(2*Pi^2) = 0.548072... . - Amiram Eldar, Oct 13 2022

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A225925 G.f.: exp( Sum_{n>=1} A002129(n^2)*x^n/n ), where A002129(n) is the excess of sum of odd divisors of n over sum of even divisors of n.

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A224339 Absolute difference between sum of odd divisors of n^2 and sum of even divisors of n^2.

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