A350166 - OEIS

%I #18 Feb 02 2024 02:39:35

%S 1,3,5,9,15,19,25,33,40,52,62,70,84,96,108,124,142,156,174,198,210,

%T 230,252,268,299,327,347,371,401,425,455,487,507,543,579,607,645,681,

%U 709,757,799,823,865,905,947,991,1037,1069,1112,1174,1210,1266,1320,1360,1420

%N Partial sums of A050469.

%H Winston de Greef, <a href="/A350166/b350166.txt">Table of n, a(n) for n = 1..10000</a>

%F a(n) = Sum_{k=1..n} Sum_{d|k} A101455(k/d) * d = Sum_{k=1..n} A050469(k).

%F G.f.: (1/(1 - x)) * Sum_{k>=1} k * x^k/(1 + x^(2*k)).

%t f[2, e_] := 2^e; f[p_, e_] := If[Mod[p, 4] == 1, (p^(e + 1) - 1)/(p - 1), (p^(e + 1) + (-1)^e)/(p + 1)]; s[1] = 1; s[n_] := Times @@ f @@@ FactorInteger[n]; Accumulate @ Array[s, 50] (* _Amiram Eldar_, Dec 18 2021 *)

%o (PARI) a(n) = sum(k=1, n, sumdiv(k, d, kronecker(-4, k/d)*d));

%o (PARI) my(N=66, x='x+O('x^N)); Vec(sum(k=1, N, k*x^k/(1+x^(2*k)))/(1-x))

%Y Cf. A050469, A101455, A350146.

%K nonn

%O 1,2

%A _Seiichi Manyama_, Dec 18 2021