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A228274 a(n) = Sum_{d|n, n/d odd} n * d.

%I A228274 #20 Nov 30 2022 08:34:12
%S A228274 1,4,12,16,30,48,56,64,117,120,132,192,182,224,360,256,306,468,380,
%T A228274 480,672,528,552,768,775,728,1080,896,870,1440,992,1024,1584,1224,
%U A228274 1680,1872,1406,1520,2184,1920,1722,2688,1892,2112,3510,2208,2256,3072,2793,3100
%N A228274 a(n) = Sum_{d|n, n/d odd} n * d.
%H A228274 G. C. Greubel, <a href="/A228274/b228274.txt">Table of n, a(n) for n = 1..10000</a>
%H A228274 M. A. Basoco, <a href="https://doi.org/10.1090/S0002-9904-1943-07907-4">On the Fourier developments of a certain class of theta quotients</a>, Bull. Amer. Math. Soc. 49 (1943), 299-306.
%F A228274 Multiplicative with a(2^e) = 4^e, a(p^e) = p^e * (p^(e+1) - 1) / (p - 1) if p>2.
%F A228274 G.f.: Sum_{k>0} k^2 * (x^k + x^(3*k)) / (1 - x^(2*k))^2. [see Basoco (1943) bottom page 305]
%F A228274 G.f.: Sum_{k>0} k^2 * (3 - (-1)^k)/4 * x^k / (1 - x^k)^2.
%F A228274 G.f.: Sum_{k>0 odd} k * (x^k + x^(2*k)) / (1 - x^k)^3.
%F A228274 a(n) = n * A002131(n). a(2*n) = 4 * a(n).
%F A228274 a(n) = A007331(n) - 4 * Sum_{k>0} A002131(k) * A002131(n-k). [see Basoco (1943) page 305 equation (9)]
%F A228274 Sum_{k=1..n} a(k) ~ c * n^3, where c = Pi^2/24 = 0.411233... (A222171). - _Amiram Eldar_, Nov 30 2022
%e A228274 G.f. = x + 4*x^2 + 12*x^3 + 16*x^4 + 30*x^5 + 48*x^6 + 56*x^7 + 64*x^8 + ...
%e A228274 a(6) = 48 = 6 * (2 + 6). a(9) = 117 = 9 * (1 + 3 + 9). a(10) = 120 = 10 * (2 + 10).
%t A228274 A228274[n_] := If[ n < 1, 0, n Sum[ d Mod[n / d, 2], {d, Divisors @ n}]]; Table[A228274[n], {n, 50}]
%o A228274 (PARI) {a(n) = if( n<1, 0, n * sumdiv(n, d, d * (n/d % 2)))};
%Y A228274 Cf. A002131, A007331, A222171.
%K A228274 nonn,mult
%O A228274 1,2
%A A228274 _Michael Somos_, Aug 19 2013